I have an oak threshold on an outdoor door and it faces south. The contractor brushed on some water based varnish-stuff that peeled off within about 2 months leaving bare nekkid and now-bleached wood. I think I want marine varnish, but when I went to a website from my google hit, there were too many choices and I got confused. I am in Indianapolis -- the largest US city on non-navigable water -- I don't know anything about boats and this website catered to sailors!
I would try going to your local Home Depot or Lowes--I have an extra heavy duty outdoor varnish and that's where I got it. I don't think mine is marine varnish but I remember they have that there too, but reading the description on the one I bought I felt good that it was going to give similar performance and I think it sounded less hazardous than the marine stuff. Sorry I don't have the container anymore or I would give you a brand name!
Diamond Varathane makes a high quality low V.O.C. type of urethane, but any thing that says 'spar varnish' will work.
In case you don't know, v.o.c. stands for volatile organic compounds, which we all want to avoid breathing. You might give that threshold a light sanding, and obviously, any staining should be done before you varnish or urethane it.