Homomorphic encryption is a category of systems; some implementations might be weak, and others might be strong, but it doesn't make sense to talk of the entire category as "weak" or cryptanalyzable.
Partially homomorphic cryptosystems (which used to be called just "homomorphic" before "fully homomorphic" cryptosystems were discovered) have been used in crypto for a while, including, as Neal points out, in my voting system, Helios. In these systems, you can perform one operation, either addition OR multiplication, under the covers of encryption. That lets you do interesting things, like counting individual votes and only decrypting the tally.
Now, when I say "don't use Helios for public-office elections," it's not because of any weakness in homomorphic encryption. That's the strongest part of the system. The problem with online voting is that your desktop client could be compromised by malware, thereby changing your vote before it is encrypted. The homomorphic tallying portion is quite secure, and there are no known attacks against it.
Boneh, Goh, and Nissim designed a more homomorphic cryptosystem in 2005, where you could do any number of additions, followed by one multiplication, followed by any number of additions, before decrypting. That enabled more interesting applications, e.g. my work on Public Mixing (also applicable to voting), where you can shuffle a set of encrypted values in a public operation, without revealing in what order you shuffled them (pretty crazy, when you think about it.)
Fully hommomorphic cryptosystems, where you can do arbitrary additions and multiplications, were thought to be impossible until Gentry's work a couple of years ago. What's meaningful about this category of cryptosystem is that you could fully outsource any computation to the cloud without ever revealing plaintext data. For example, if you wanted to perform a full text search of the word "cryptography" on a corpus of text, you could encrypt the corpus, encrypt the word "cryptography", and ship that to another party who would perform the full-text search on fully encrypted data, and return to you the encrypted result, which you could then decrypt to get the answer. The system that does the computation would known nothing about the corpus or the search query. Pretty amazing.
But of course, this only makes sense if the process of encrypting, and the process of performing homomorphic operations, is still cheaper on the cloud than doing it yourself in plaintext on your local machine. We're very, very far from that. That said, cryptosystems only get better with time, so maybe we'll see generic homomorphic computation become useful in a few years.
In the meantime, there are probably plenty of specific problems -- not generic computation -- that can be outsourced more securely thanks to homomorphic technology.