Yes, finishing any wood exposed to the weather is a good idea. You have several choices;
1- Apply a suitable film forming clear finish. There are several on the market that are suited for exterior use, both oil base and water base. This could be spar varnish or acrylic urethane.
2- Apply a penetrating oil, like linseed oil. There are many types of penetrating clear finishes that could work. Messmer's, Sikens and Cabets are just a few with every manufacture offering their own types of products.
3- Apply a semi-transparent, semi-solid or solid wood stain.
To determine which option is best for you visit your local paint store and ask for recommendations.