Sinew and hide glue is always going to be susceptible to moisture problems in humidity or rain. The best you can do is slow the effect by using a good finish on top. No finish is going to be waterproof, but it will be some help.
Ed Scott of Owl Bows claims to have "water proof" sinew backed bows. He told me that he uses TBII for application of sinew and then TBIII as a top coat or to apply skins over the sinew. Of course, he lives in AZ so what would he know about humidity.