if you're going down on Sunday, there is a huge boot sale in the station parking lot. Its a bit hit and miss, but I've found some really interesting furniture there in the past.
Also, first Sunday of every month, there is an organic farmers market in Hove Town Hall.
Other things to do:
Catch the number 77 bus (the open top bus) up to Devil's Dyke. Its not the beach, but it is gorgeous countryside and well worth the £2.40 bus fare.
If you're hanging out on the beach and want to get away from the tourists, head towards the West Pier. Just underneath the West Pier is a little market area filled with stalls selling incense, homemade clothes, etc. The beach there is much nicer and much less crowded.
have fun!
-prakriti