1X TAE will not last long, say approx. 7-10 days depending on the number of times you open the bottle to the atmosphere. However, I prepare 50X TAE which lasts for 2-3 months.
Please note while preparing TAE use sterile double distilled water. Do not autoclave TAE buffer.
50X TAE is very stable, we make it 1-4 times a year (it depends on how many gels we are running). It hasn't gone moldy/bad yet.
The 1X TAE can eventually grow microbes, if it starts to smell odd it likely has mold. If you use a clean container and keep the lid shut it will keep for weeks.
The 1X TAE in the gel running rigs tends to go bad faster than in a storage bottle. I would suggest changing it daily (or if you run infrequent gels just store the rigs dry and empty)