Probably the most important thing is to make sure the sinew is dry before you put on the finish. I think a waterproof finish is best....pitch, beeswax, or any type of oil that dries hard (linseed, tung, walnut) or a mixture of these. Tru-oil certainly qualifies.
I've tried applying a finely ground dry pigment (or powder) to the sinew. I've also "painted" on an alum/water solution over the sinew to make it less sticky and more water resistant. Both of these things have worked for me....if I choose not to put a finish over the sinew.