There are, in my opinion, at least three key options regarding security:
1) My old-school opinion is, a hosting service that uses dedicated hardware kept in physically secure server farms. It is not a low cost option though.
2) More affordable is a hosting service that uses virtual shared hosts on dedicated hardware. This is probably about as secure as option one, but it won't be as configurable and you'll have to go through special software managed by the hosting company to do some kinds of system administration. So, not quite as safe.
3) Least secure, and least reliable, are servers implemented "in the cloud" such as virtual machines form Microsoft or Amazon. I consider them less secure because of the large blanket of configuration and security software necessary to manage them. And that these services have more trouble is a matter of record, which is why when you use them, they always warn you to build a backup infrastructure NOT in the cloud in case of a major breakdown. The cloud turns out, in most cases, also to be the most expensive option, which is hardly surprising. Because what you really gain from the cloud is instant re-configurability, and ease of upgrades/migration--things like that.
Of course, the performance record of an individual company matters. The biggest problem I encountered over the years was companies going out of business of being bought up and radically changing their pricing or support policies. So I look for historical stability, support, security, fiber pipelines, pricing and a number of other features. And, I don't use the cloud, period.
I am not including any specific company recommendation because you did not specify which platform or operating system you need. There are some companies good for Microsoft hosting, others for Linux, and still others for special needs.