Switching from 1024-bit to 2048-bit means that the operations with the asymmetric private key will take about 8 times as much CPU. A "normal PC" core can do about 1000 1024-bit RSA operations per second, down to about 125 per second for 2048-bit RSA. Note that RSA is used only at the start of the SSL connection, not afterwards; moreover, in SSL, there is an abbreviated handshake by which the client and the server reuse the cryptographic exchange from a previous handshake. Web browsers and Web servers do that a lot, so the cost of RSA in SSL is really for new clients (i.e. clients who have not connected since the last time they closed and reopened their browser). It thus takes a hundred or so new clients per second to actually see performance issues related to the key size on the server. Not every Web server out there can boast such a context.
I encourage to make measures in an actual realistic context. Microbenchmarks ("how many decryptions per second ?") are not always good predictors.
As for security, 1024-bit RSA is, as yet, unbroken. Current breakage record is RSA-768. Some smart people have tried to equate RSA Key length with symmetric key robustness; see this site for details. The comparison is not perfect because breaking big RSA keys involves some operations which use an awful lot of very fast RAM, quite unlike exhaustive search on a symmetric key (an embarrassingly parallel task which has no need for RAM). Yet, 1024-bit RSA is somewhat equivalent to a 77 to 80-bit symmetric key. That's not enough for the paranoid, but still beyond that which has been demonstrated to be breakable with currently deployed technology.
NIST recommends using 2048-bit RSA keys right now (since 2010). If you use a 1024-bit RSA key, your site won't be broken that way (when it is hacked, it will be through another channel than upfront cryptanalysis); however, your customers might conceive the idea that by using an unfashionably small key, you don't really care about them. It is more about public relations than security...
If you really have actual performance trouble with 2048-bit RSA keys (I don't believe it, but hey, weird things happen), I encourage you to investigate elliptic curves. An ECDHE_ECDSA cipher suite with the P-256 NIST curve will be faster than RSA-1024, and as strong as (arguably stronger than) RSA-2048.