I think that to make a transmissable ("contagious") antiviral vector, you would need to design a virus that can infect people and that codes for something like an antiviral drug compound, which targets specific viral proteins.
As easy as that was to write down in one sentence, I think that execution would be extremely challenging.
Making the anti-virus out of another virus seems necessary to me, as there is little alternative. You would need something capable of efficiently spreading from host to host, that can generate many copies of its therapeutic payload and that is relatively simple to engineer. A virus is the only thing that I can think of that fits the bill.
You could theoretically engineer an antiviral virus that is mostly non-pathogenic to other people, although this wouldn't be perfect. Even if it didn't code for specifically pathogenic proteins, it is a foreign molecule that is likely to raise an immune response, risks integrating into an important location in the host genome and would divert cellular energy to production of its proteins and away from other cellular processes.
Say that you get this to work, that it's minimally harmful to its host and can effectively target the more pathogenic virus, against which you've designed it. What happens when the job is done? Is it still in the host body, pumping out more copies of itself and/or its antiviral proteins? If so, then it could end up causing more damage to the host in the long run, as it uses host resources to fill cells with no-longer-helpful foreign molecules.
Given more thought or time (than I have), you could probably image more ways to tweak the design of this theoretical anti-virus, but as a brief answer, I think that the above considerations can give you a quick idea of what it would take to bring this system from the realm of science fiction to that of hard science. :-)