Our cruise ship docks in Zeebrugge, Belgium, for 1 day. We are not interested in the ship/shore expensive excursions. Can anyone recommend an alternative to see the sights on our own?

There are lots of excursions you can easily do on your own. Here are a few suggestions:
Brugge: 16 min. by train from Zeebrugge Dorp
Oostende (Ostend): 15 min. by train or 45 min. by coastal tram (there's a tram stop close to the harbour)
Knokke-Heist: another - more upscale - coastal town with a large nature reserve, 15 min. by coastal tram

Hi Halbaby, I would highly recommend Brugge over the seaside towns. There isn't much to them, while there are many sights to see in Brugge and it is small enough to spend a single day in. It depends on what your interests are, but Oostende and Zeebrugge don't have much to offer other than beaches (which are pretty cold except in summer) and tourist-trap restaurants. Definitely hop on a train to Brugge, it 's a lovely place to walk around. There are really beautiful paths around the old town.

Halbaby: mp14 is right on. In addition, the area is very industrial; not pretty. Brugge is delightful.

<<the area is very industrial; not pretty>>
Really? Where do you see that industry in Knokke? Or even in Oostende?
Halbaby didn't tell us when (s)he was visiting here, that's why I suggested both seaside towns.
Of course Brugge would be my first choice, too but even in Oostende there's enough to do to keep one busy (it is our 'Queen of Seaside Towns, after all!!): museums, art galleries, not to forget some great fish restaurants at the Visserskaai!

Halbaby, definitely don't stay in Zeebrugge. It's an unattractive (industrial) towns and there's really nothing to do (unless having lunch/dinner at Maison Vandamme or 't Molentje).