There needs to be incentive that's primarily beneficial to PvP while also not resulting in the "rich get richer" that someone in the comments also mentioned.

1) There's a difference between "PvP happens to drop guns that can be really good in PvE" and "these PvP weapons are the PvE meta and not having them will make you feel behind" (like Recluse and Mountaintop). The new Trials Linear Fusion is a great example of what to do right, as it has some top tier PvE potential but not so far above everything else that PvE players feel required to get it.

2) I think some kind of guarantees of good armor stats or weapons with multiple perk selections are good. Locking rolls exclusive to competitive that will shred the crucible means that the first people to get it will dominate, making it harder for people after them to get it and discourage people from trying.