As others have mentioned, there could be some interference. Most likely there is not a safety issue here if you are running parallel to standard plastic-sheathed NM / romex wire, but you may see reduced speeds. It's certainly not against code. You should at least use CAT 5e over CAT 5, as it is shielded better and can almost work at Gigabit speeds. CAT 6a would be your best bet, but probably a little overkill for you. When I built, I ran 5e in my house, and tried to keep it away from the the electrical as much as I could. There was some crossover and parallel runs, but it was kept at a min and I have been getting near gigabit speeds.
Note, manufactures have been using the term CAT6e, which does not appear to be a true spec. Seems like they are loosely using that as CAT6 with extra shielding. CAT6a on wikipedia.
Here is an article I found:
differences between cat5 cat5e cat6 and cat6e cables