Well of course you have to vent the closet somehow like you would any grow room. A 400 watt digital system doesn't get too hot, like say 600+ watts would. Especially if you have an air-cooled reflector. Even if it did get 100 degrees in the closet nothing should catch on fire, unless you're keeping highly flammable materials in your closet\grow room... which you really shouldn't be.
A regular household fan isn't going to cut it. It will just be circulating the same hot air. Even if you don't have an air cooled reflector, you should still definitely vent the grow room somehow. Either make two holes; one high to the ceiling and one low to the ground, to vent passively, or make a hole in the ceiling (with a circular saw) and set up some ducting and a high output inline fan if this is convenient (such as if there is an attic above the closet).