Bitcoin wallet not Connecting

My client stopped synchronizing, how can I access my wallet?

It's not your wallet you need to "get current"; it's the blockchain. However, the standard client does need an internet connection for many of its functions.

Firstly, you want to ensure you've got a backup of your wallet.dat (preferably encrypted) file, which contains the private keys you need to recover your funds. If it was me, I'd make 2 copies and store them on different machines in different locations.

Once you've done that, you need to get your client app up and running. This is likely to take some further analysis, but to start with can you please state the version of the client you're trying to use, and whether or not you are using a proxy server for your internet connection.

Note also that the blockchain is available for download separately; you'd save a lot of time doing it that way as synchronization can take hours (it's a disk-intensive process). See details here: Can I download the whole block chain from somewhere?