Even a tiny amount of endonuclease is enough to perform a single nick and convert the covalently close, supercoiled form of a plasmid into the open, relaxed form,. The main advice would be to work with DNase-free (autoclaved) solutions and DNase-free plasticware. Also, avoid acidity by resuspending the DNA into TE buffer (10 mM Tris-HCl pH 8, 1 mM EDTA). EDTA will chelate traces of Mg that are required for most nucleases; however, once you perform your assay, you can add sufficient Mg (e.g. 10 mM) to overtitrate the EDTA present in TE buffer.

Normally the supercoiled form should be quite stable. I have had plasmid preps that were many months old that were still supercoiled. So it could be something is not quite right with your procedure or your final buffer. It could also be a strain dependent problem. If your E. coli strain is EndA+ you are highly likely to have some cross contaminating endonuclease in the prep.

So check to be sure you have EDTA in your storage buffer as mentioned by Pierre Béguin and what strain you are isolating the plasmid from.

supercoiled form of pDNA is very stable (many months at -20C). After a typical purification with a well established technology (eg Invitrogens PureLink or Qiagen kit) you have most of your plasmid in this form, with a few minor bands seen on a gel (typically totaling < 10%). If your supercoil fraction decreases rapidly over time- these are nucleases or chemical impurities causing nicks. Pay attention to your plasmid purification kit and elution/storage buffer