[meowface]

It depends.

The Dogecoin client is based off the Litecoin client. The Litecoin client is based off the official Bitcoin client.

Odds are, if a remotely exploitable bug is present in Dogecoin's network service, it will probably be present in Bitcoin's too, and every altcoin that is a direct fork of Bitcoin.

Cryptocurrencies require you to establish P2P connections with other people on the Internet, so if you want the network to thrive you do have to run it on a public network. There is always going to be a risk with any service you run, but Bitcoin's code has been audited by security professionals on quite a few occasions, so the chances of a critical bug that can be remotely exploited against someone simply running the client are fairly low. There's probably a higher chance of there being a critical bug in uTorrent.