Everything for finishing wood gives off fumes. You need to let the cage air out for enough time to get the fumes out. Finishes only stop smelling when the finish is fully cured. Some finishes like water-based paint cure quickly; other finishes like oil-based polyurethane take longer.
So, pick what you want, then let the cage air out enough time (this could be weeks!)
Here are some choices:
Paint (first primer, then use water-based gloss or semi-gloss acrylic)
Polyurethane (water-based cures quickly, oil longer)
Epoxy resin (as suggested by others on this forum, look for Envirotex)
In my experience, I think paint is the best because you can do colors, and easiest because it is repairable/replaceable should the finish get damaged. It isn't the toughest though. Epoxy is the toughest but is pricey and a bee-aa-tch to work with. Poly is what every Joe Schmoe thinks is the ultimate answer to all wood cage sealing applications (then finds out it has limitations just like everything else and that paint is probably just as effective).