I've read that it's never safe to use the same bitcoin address more than once because there is a small chance that someone could solve for your private keys (though I don't have a solid-enough background in encryption to understand all of the implications of this concept). In any case, even if you did generate a different address for every incoming transaction, what would stop an attacker from simply sending many transactions to the first address they see in an attempt to crack your private key?
So there are attacks where if you sign enough times with the same private key, the attacker, with access to a side-channel can figure out your key.
So if you were signing with your private key a lot of times, this would make your private key slightly less secure. If you're receiving bitcoins to your address, you're not signing with your private key, so this isn't an issue.
The more important consideration here is that you lose privacy with address reuse.
Several cryptographic schemes have such kind of flaw, that is, if an attacker intercepts several messages from one single private key source, it becomes easier to decipher the message. During the World War II this were done several times by the allies in order to break the Nazis and the Japaneses codes.
Fortunately, bitcoin rely on cryptographic schemes (mainly SHA256) that doesn't have any know flaw (unless brute force) yet. This means that, if you don't provide your private key to no one else, you can do as many transactions you like with one single address account that an attacker would not be better of to break your account.
It does make your address slightly less secure once you send from it. If the address is only used to receive Bitcoin though, it is secure, and there is no significant risk to your private keys.
EDIT Added: Sending lets the world know the public address. This increases risk (albeit to insignificant levels) but still IMO many times higher risk than not disclosing this information at all, because now any attacker is closer to finding the private key.
So as just one reason (there are more)... given that there is clearly non-zero risk that technology could be developed to compute private keys more easily from public addresses, publicizing public addresses increases risk. See link below.