First of all, for any red wine or port, I would suggest a vacuum stopper, sort of like this....
...instead of a plain stopper. You can get these almost anywhere.
Ports are a little different in that the higher alcohol and sugar content help to preserve them longer than a typical red wine might last. I've had them keep for several weeks (I was very surprised) but I wouldn't recommend it. One little taste will tell quickly if it is OK, but no worries about getting sick. It'll just get bitter.
The vacuum stoppers work because oxidation is only your friend for a short while (hence the need sometimes to let a wine 'breathe'). After that, it is definitely the enemy. A vacuum stopper removes much of the air, and thereby slows the rate of oxidation significantly. I've actually managed to keep some red wines a few days with the vacuum stopper where they wouldn't keep for more than a day without it.
The stoppers are cheap and well worth the money.