Puerto Rico may be losing some of its tourists, but I doubt they are losing them to the DR.
First, most tourists to PR don't even stay in hotels and I doubt most stay in villas either. It seems to point towards a tourism largely based on Puerto Ricans and descendants on the mainland visiting their homeland. Not that they can't visit the DR, but the DR certainly lacks, at least for most, the one thing PR has and that is family members.
Second, of the tourists that do stay in hotels, average stays are of 2 or 3 nights. It appears that most non-Puerto Rican tourists tend to visit PR due to some cruise they plan to take out of San Juan. I can't think of another reason why average hotel stays are so low in PR. They visit the island to get on a cruise and in the mean time, visit the sights within the San Juan area. Since the DR is not a major cruise home port, despite the development of Sans Souci in Santo Domingo, I doubt the cruise traffic is shifting out of PR towards the DR. It could very well be shifting from PR to Miami.
Third, most tourist hotels in PR are lined up on the Condado and Isla Verde beaches in San Juan. There are a few megaresorts outside of the San Juan area, such as the Rio Mar complex I think is in Dorado and a few other complexes out in the Fajardo area on the eastern tip. But overall, these complexes are few and far between. The bulk of PR tourism takes place in the two San Juan beaches already mentioned and, while nice beaches, they are not true competition for beaches like Punta Cana or even Playa Dorada. Even in PR, you will find nicer beaches away from San Juan. My point is that if beach competition was the major draw of tourists from PR to DR, this shift would had happened a long time ago, with or without crime in PR.
And lastly, Dominican tourism functions and targets a completely different market segment then does PR tourism. Its ironic, but the two systems are almost incompatible. Dominican tourism has more in common with tourism in Jamaica, the Bahamas, but especially Cuba. Also the Cancun-Mexican Riviera in Mexico is a strong rival. But I think Cuba is the biggest true competitor, mostly because its an All-Inclusive model that depends almost entirely on the European and Canadian market; similar to the DR except we have the American market as well. PR, Cancun (heck, all of Mexico), Jamaica, Bahamas, Aruba etc are dominated by Americans to a degree not seen in the DR and, minus PR, they are predominantly all-inclusive destinations as well. The rest of the Caribbean I think is a non-issue since they are not predominantly all-inclusive and in so many of the smaller islands, the dominanting tourist nationality tends to be the former colonialists (the French dominate tourism in Martinique and Guadaloupe, to give one example.)
PR and DR tourism are like water and vinegar, IMO.
Having said that, I do remember having read a few weeks ago an article in which some sort of cooperation is going to be tried between the two islands. Its suppose to be some sort of marketing ploy to present both islands as one destination. I don't think it will work mostly for reasons already expressed here, but we'll see...