There are several factors at finding a secure and reliable web hosting provider, and it starts at the core infrastructure.
- What architecture do they have in place for the web servers, database servers, dns servers, etc? Is it a redundant setup so if there is a denial of service attack or you get excessive traffic your site will still be online?
- Is your website and database server hosted on the same machine? Will your whole site go down if one piece goes down for maintenance? It helps to have redundancy so that proper maintenance could be done in a secure and reliable manner.
- What server providers do they use? Is it only one or do they have redundant links? (Are you seeing a theme yet?)
- What performance metrics can be seen and can you determine their historical performance? Is it a stable provider or are they having trouble all the time?
- Are they a reseller to another service provider? When it comes to architecture, you might have to dig deeper into the actual provider as the reseller might not know themselves, which is an indicator that they are not on top of their game.