1) In my laboratory we run both SDS-PAGE and submerged Western blot transfers at a constant current, because then we can walk away from them and return at a specific time which is known. Some labs run SDS-PAGE at a low voltage until the protein front (containing the Bromophenol blue) crosses over from the stacking gel to the separating gel, and then they increase the voltage. I have done so in the past in the lab where I did my PhD, it works fine. The resistance of the SDS-PAGE changes during the running of the gel, so it occurs that with a power supply set at constant current, the voltage automatically increases to compensate for the increased resistance that the gel develops during the course of the running. So that means I don't have to come back and change the settings, the power supply adjusts the voltage automatically. The best current for nice sharp bands on SDS-PAGE depends on the size of your gel (how wide is it, how high is it) as well as the thickness of spacers and combs that determine the distance between the two glass plates). In my lab we have home-build equipment because we find that commercial ones don't have enough slots. So we had to find out empirically what the best running current is. Commercial equipment should come with instructions and recommended running conditions. But whatever the method, you should run your gel slowly at the beginning so that the stacking gel can do its job. It is clear why, all the protein bands line up together in one sharp front and when you rush this step it yields smiling bands with tails. Slow is always better. It also helps if your slots are not filled up with sample higher than their width. The stacking gel stacks, but it doesn't do miracles.

2) Western blot transfers are always done at constant current and the resistance is more or less constant (if it is cooled). If it is a totally immersed transfer, the conditions are always the same regardless of the size of your gel and nitrocellulose membrane. If you do semi-dry blotting however, then the resistance depends on the wetted area, and depending on size you may need to calculate and change the conditions. Semi-dry blotting saves on transfer buffer (Tris, glycine and methanol) but you run the risk of uneven blotting. Totally immersed blotting is safer, unless you made fresh buffer and solubilised too much air in the event of rapid mixing, which then forms bubbles when the liquid warms up during the transfer. Always make transfer buffer well in advance and by slow stirring with a magnet stirrer to avoid excess.  

1) In my laboratory we run both SDS-PAGE and submerged Western blot transfers at a constant current, because then we can walk away from them and return at a specific time which is known. Some labs run SDS-PAGE at a low voltage until the protein front (containing the Bromophenol blue) crosses over from the stacking gel to the separating gel, and then they increase the voltage. I have done so in the past in the lab where I did my PhD, it works fine. The resistance of the SDS-PAGE changes during the running of the gel, so it occurs that with a power supply set at constant current, the voltage automatically increases to compensate for the increased resistance that the gel develops during the course of the running. So that means I don't have to come back and change the settings, the power supply adjusts the voltage automatically. The best current for nice sharp bands on SDS-PAGE depends on the size of your gel (how wide is it, how high is it) as well as the thickness of spacers and combs that determine the distance between the two glass plates). In my lab we have home-build equipment because we find that commercial ones don't have enough slots. So we had to find out empirically what the best running current is. Commercial equipment should come with instructions and recommended running conditions. But whatever the method, you should run your gel slowly at the beginning so that the stacking gel can do its job. It is clear why, all the protein bands line up together in one sharp front and when you rush this step it yields smiling bands with tails. Slow is always better. It also helps if your slots are not filled up with sample higher than their width. The stacking gel stacks, but it doesn't do miracles.

2) Western blot transfers are always done at constant current and the resistance is more or less constant (if it is cooled). If it is a totally immersed transfer, the conditions are always the same regardless of the size of your gel and nitrocellulose membrane. If you do semi-dry blotting however, then the resistance depends on the wetted area, and depending on size you may need to calculate and change the conditions. Semi-dry blotting saves on transfer buffer (Tris, glycine and methanol) but you run the risk of uneven blotting. Totally immersed blotting is safer, unless you made fresh buffer and solubilised too much air in the event of rapid mixing, which then forms bubbles when the liquid warms up during the transfer. Always make transfer buffer well in advance and by slow stirring with a magnet stirrer to avoid excess.  

Hi

The reason why macro-molecules migrate in the gel is the difference of potentials (generally referred to as 'voltage'). The higher is the voltage the faster is the migration, and this relationship is linear. The voltage is the controlling parameter here, the actual "driving force" of the whole process. On the contrary, the current depends on both the voltage and the resistance and tends to vary because the resistance increases gradually during electrophoresis.

Apart from an explanation of Ohms law describing the relationship between voltage current and resistance, the most relevant details are to be found in Jurgens answer

I will to what he said by saying that when resolving different sized proteins by PAGE the use of a constant voltage or current is not so critical

The more crictical part oif the procedure is subsequent transfer of proteins to NC or Hybond, i.e. blotting. Genrally speaking you run at a constant and usually low current to minimise the generation of heat as this can complicate blotting efficiency and lead to the gel sticking to the NC (or hybond) if the current is too high and therefore too much heat is being generated. To elaborate in terms of Ohms law, the higher the voltage the higher the current for a given resistance (which admittedly changes during blotting owing to electrolytic break down of your blotting buffer): this leads to more effective and rapid transfer but generation of more heat and associated problems (described). However, even this can be controlled at higher currents for more rapid transfer by simply immersing the blotting rig in ice or performing blotting in a cold room (4C)

Very good answers to your question have been given above. The thing is: V= I*R. Even though you set up a constant Voltage or Current at the beginning, at some point, it'll change and become non-constant. So, you don't really need to worry about it, just go ahead!  

I agree with the above. Electrolytic breakdown of running and transfer buffer leads to increased resistance and lower current for a given voltage

However, you should still concern your self with production of heat during blotting at higher currents unless as mentioned you keep things cool  :-)

Please remember that not only the transfer generates heat, also the running of the SDS-PAGE does. We therefore run both the gels and the transfers under controlled temperature by submerging into a sludge of  ice and water.