There's a lot of really good options at that price range, I'd take a trip down to the guitar store and try a few out to see what you like! Are there any bass players with tone you really like? Any basses you just love the look of? I'd start there.
The Sire basses are an excellent, modern take on your usual P-bass and J-bass styles. If I remember correctly, most of them have slightly narrower nut widths then their Fender counterparts (great if you like a more narrow neck).
The Ibanez basses absolutely flood the market with variety at this price point, so it's tough to make a general statement about them. The quality is good, and you can get just about any body shape or pickup orientation you can imagine.
Yamaha also makes a version of the BB bass below a thousand bucks. I've never tried it, but if it's as good as a the four string version, then it's worth checking out.
Schecter has a lot of offerings too, and don't be deceived by the aesthetic - they're really versatile basses, not just for metal!
Same comment for Schecter applies for ESP LTD (but in general, I think Schecter quality is better).
There's a Fender Player series jazz bass too. I've heard comments from lots of people saying Fender is still dialing in their five string models, but I own one and absolutely love it.
The Sterling Music Mans are a classic, if that's the tone you're after - this is peak Leo Fender as far as I'm concerned.