1) Tourist tax on all hotel rooms, including AirBnB. Initially a fairly small amount. https://www.theguardian.com/uk-news/2018/jun/27/edinburgh-le...
2) For whole-unit owner-not-present lets, require planning permission for change of use. You wouldn't be allowed to run a woodworking business in your apartment that inflicted noise on your neighbours, so the same should apply to turning it into a hotel.
3) Keep upgrading the transport infrastructure. The newly rebuilt Border railway is a victim of its own instant success - someone should start the process of widening it to two tracks immediately so it'll be ready in 20 years. And so on.
(Before someone says "build houses", there is a lot of family home construction going on in the outskirts. It's the historic centre that's under consideration, and "demolish the existing 4-6 story historic dwellings and replace them with much taller residential buildings" is an absolute non-starter)