First, do not mix up the words "wallet" and "address". A wallet is a file or computer program which stores the private keys corresponding to one or more Bitcoin addresses. Sharing the address is very different from sharing the private keys.

Sharing the address is not a security risk, in the sense that knowing the address does not give anyone the ability to spend your bitcoins. They would have to know the private key. Indeed, since Bitcoin uses a public transaction register, you can assume everyone in the world already knows the address 123abc exists, and knows how many bitcoins are credited to that address.

However, sharing the address can be a privacy risk. Everybody knows that the address 123abc exists; what they don't know is who it belongs to. If you post the address on your personal website, or from an account that can be connected to you, now they do know that.

For instance, suppose you made a transaction from that address to an address that's known to belong to EmbarrassingSexToys.com. The transaction is public, so everyone can see that it was made. If you hadn't posted your address on the Internet, then everybody knows that somebody bought an embarrassing sex toy, but nobody knows who it was. If you did post your address, then everyone knows that Adnan bought an embarrassing sex toy.