Install air admittance valve to solve musty smell (sewer gas)?
The washing machine sends water into that drain under quite a bit of pressure and you want air moving freely through the pipe. You would be better off if you could run a new vent pipe up that wall and out the roof, or you could go as high as possible then out through the wall, presuming it isn't too close to a door or window that opens (since sewer gases exit the vent pipe on a regular basis).

Barring that, however, you should certainly put an AAV on this. I've had a plumbing inspector approve exactly this kind of setup for a washing machine drain on a remodel. The inspector required 4 feet of vertical pipe between the AAV and the wye that tied it into the drain pipe, though.

Your diagram may not be intended to be to scale, but I'd say you definitely want the AAV higher than what you're showing. I certainly wouldn't try to get away with putting the AAV under the sink.