How you finish OSB depends on its internal adhesive. If you use AdvanTech, for instance, which uses an epoxy resin, you can stain it or varnish it - but you can't use any sort of latex- or water-based primer or stain. You'll have to read the manufacturer's installation instructions.
In general, you can stain OSB with an oil-based interior or exterior stain. You can also apply a polyacrylic, polyurethane, or spar urethane finish. Spar urethane might last longer since the OSB will change size with temperature. So long as you don't mix incompatible materials (see above paragraph), it could work. If you're going this route, using a solid or semi-solid exterior oil-based stain and coat with 4-5 coats of spar urethane. You'll need the UV and moisture resistance. Transparent stains are pretty, I know, but the more transparent the stain, the less the wood is protected. It won't last 30 years, for sure, but it'll work.
Most OSB's respond better to oil-based primers and stains. Several "exterior" OSB's also have a wax coating which can interfere with adhesion.
The problem isn't chemical compatibility. It's durability. Even the best OSB panels are rated Exposure 1, and NOT Exterior, and certainly not marine. On a cheap floor? No big deal. But they also epoxy or literally POUR polyurethane onto the OSB to give it that sleek, shiny look.
Anyone ever try using aniline dye on OSB??? Wonder how that would work ...
Painting the OSB technically gives superior moisture and UV resistance, but it comes with another problem: telegraphing. OSB doesn't sand well, so you'll have to sand, sand, then prime, prime, and then paint, paint and paint. OSB sucks up paint like crazy. Some people use masonry "block filler" to fill it out to get a smooth surface.
Something else to consider: If you drill through the OSB, you'll have to waterproof all your fastener holes with rubber washers or bed your fasteners with silicone or some such. Else water will wick into the holes and get behind your protected surface and rot the OSB from the inside out.
You can clear-coat epoxy OSB, yes, but that cuts into the "cheap" part of it ...
Really, it's a question of your time. Can you get an OSB exterior to work? Yeah, you can. But it'll be a 7-step process before you're through. If you want the special look of finished OSB - which some people love - then it's totally worth it. Some people have even built OSB boats. But if you're just looking to save money ...

Well, if you like the look of old gray barn wood, why not use it? Lots of people will let you haul off their scrap wood for free. Just a thought!
