Are you sure it is teak? Most of the early Day Sailers had mahogany wood trim (cockpit coamings, floorboards, short seats that connect CB trunk to side seats) It is possible that teak was an option, or that a previous owner upgraded from mahogany to teak.
Your options are sand smooth and coat periodically with a good teak oil, san smooth and apply varnish (thin first coat, then build up 3-5 more coats, sand lightly each year and apply 1-2 new coats), Apply Deks Olje ("Decks Olay") oil finish (part 1 is an oil similar to teak oil, part 2 is an optional sealer that produces a varnish like finish but is easier to maintain), or finally just lightly sand to remove splinters and let the teak naturally weather. There are many different "varnishes", or different formulations and prices. Many are not true varnishes but more correctly clear-finishes.
If you decide to try any idea other than letting hte teak weather, be sure to follow the instructions fully for what ever finish you choose.