You mention "Apache" and "browsers" in your question, but the title is more general.
As Evan and others point out, the problem is all-but-sorted for HTTPS. But there are a number of other protocols that a server might encrypt, and TLS support is much poorer amongst that client base (as I found out this morning, when mandating "no SSL3" on an IMAP/S server).
So I'm afraid the answer is "it depends on what services you encrypt, and the client support for TLS amongst your user base".
Edit: yes, that was my point, though I'm glad you agree. Turning off sslv3 is done on a service-by-service basis. For example, the way to turn it off on dovecot is to put
ssl_cipher_list = ALL:!LOW:!SSLv2:!SSLv3:!EXP:!aNULL
dovecot.conf. The bigger problem is that whilst most browsers are tolerant of the loss of sslv3, clients of other services seem to be a lot less tolerant. I broke about half my users this morning when I turned that off on dovecot; Android phones running K-9 mail and Outlook on Win7 are two that I know of for sure, but I can see from my logs there were more.
Turning off SSLv3 is still not only a valid solution, it is the only solution; but it's going to hurt.
Edit 2: thanks to dave_thompson_085 for pointing out that disabling SSLv3 ciphers in dovecot disables not the just the SSLv3 protocol, but TLSv1.0 and TLSv1.1 as well, since they have no ciphers that the earlier protocol doesn't. Dovecot (at least, earlier versions, which include the one I'm running) seems to lack the ability to configure protocols rather than ciphersuites. This probably explains why doing it broke so many clients.