There is no merit in using a weaker hash function (except in some very exotic, unrelated cases, I remember a scheme that uses a weak hash as a means to prevent MITM attacks on DH). As you've already stated yourself, this only makes the attacker's life a lot easier. If you use a 32-bit hash, there cannot possibly be more than 232 passwords to check before one comes out as "valid", even if luck is totally against the attacker. That's not even the average case (which would be bad), but the best imaginable case, so this is a desaster. Of course, typical average user passwords still only have a strength of 20-30 bits, even if your hash is 128 bits, but at least you are not giving your keyspace a very small hard limit by design.
Due to the available computional power, the "only correct" way of doing passwords nowaday is to make hashing deliberately slow (rehashing, bcrypt/scrypt, you name it). Alternatively, you might not use hashing at all, but public key cryptography. This avoids the problems of compromising the database alltogether, but it has some different issues (and it's harder to get right, so I would advise you stay with hashing, which is a straightforward, well-established, well-known thing).
But it's really necessary to do something computionally much more intense than "SHA with salt", otherwise, you can almost just as well store plaintext passwords, since the effective security is zero.
If someone stealing your database is something you want to guard against (which is wise!), you must be prepared against an attacker who can literally check billions of hashes per second (a single desktop computer can do several hundred million per second, assume an attacker owns half a dozen computers, plus controls maybe 20 or 30 zombies).
Having a validation scheme that runs much slower (something on the order of several thousand hashes) means of course that you a bit of CPU time at every login, but it also means that your users have days to change their password rather than seconds when your database is stolen. Considering that it could already take a few days before you even notice, this is urgently necessary.