The are near Piazza di Spagna/Villa Borghese area is known also as “Il Tridente” (The Trident). It is one of the most upscale areas in Rome. I can’t recommend any hotels in that area because I never stay there (too expensive for my wallet).
Anyplace in the Rome Historical City center is safe.
The less safe areas in Italian cities will always be far away from the inner city. Unlike the US, in Europe city centers and inner cities are the safer areas. Suburbs and the areas far away from the city centers, the “periphery” as the Italian call them, is where the less safe areas may be. Not all periphery areas are unsafe, of course, some are also upscale, but if there are less safe areas those will generally be away from the city center. Since you, as a tourist, probably prefer to stay in the historical center, you don’t have much to worry about safety. The historical centers of Italy is where the Italian upper class lives. Those areas, which are also generally closed to car traffic (except for residents), are known as ZTL (Zona traffico Limitato, or Limited Traffic zone). ZTL, in the Italian jargon, has become synonymous with “wealthy”. So the “upper class” is often referred to as the “ZTL class”.
Now, it doesn’t matter where you are staying in Rome (or any large city anywhere in the world) you still need to watch out for thieves and pickpockets who tend to target tourists. So don’t leave your valuables unattended, like your wallet or smartphone on a restaurant table or half hanging out of your back pocket. Pickpockets a d purse snatchers are quick. So watch out and be alert, especially in crowded places (like buses or any event where crowds may form).