For blues, I would look to the Fender Jazz Bass or Precision Bass. The Jazz Bass is a little more versatile as it has two pickups, and you can blend them any way you like. The Precision Bass has a single pickup, and a deep, punchy tone.
To me, the Jazz Bass has always sounded a little slicker, and you may prefer it if you intend to play a Texas or Chicago-style blues. I’ve always thought (in this context) the Precision Bass sounds a little more woody and open, and I’d probably prefer it for more traditional Delta blues styles.
If a Fender is a little out of your price range, or if you are a beginner, check out Squier. This is a brand owned by Fender, and they make more affordable instruments based on Fender designs. They have Jazz and Precision Basses in their lineups too, and some of them are really good. I have a Squier Vintage Modified Jazz Bass, and it is outstanding, especially for the price.
Both the Jazz and Precision Bass have passive pickups, which to me sound a little more organic. If you really want a modern sound, look to a brand like Ibanez. Their SR Series basses offer a huge amount of control via the active onboard preamp, and I’ve always loved their necks. I would especially consider Ibanez if you want an extended-scale five or six-string bass.
If you have a little extra money in your pocket and aren’t keen on Fender for whatever reason, check out G&L or Lakland. Both brands have some great instruments based on classic designs, and they are a step off the beaten path for those looking for something different.