A lot of it depends on what the system does and what the consequences of defects is? Are you writing financial software? Or social networking? The likely cost of one failing are greater than the other so the QA effort varies. Similarly if it's a product (with multiple installed user bases which might be harder to patch) you need it to be more stable than if it's internal.
You also need to ask are you expecting them to do basic system testing or volume and load testing too. It sounds like you're looking at a combination of UI testing and API testing.
Tgive you a starting point, assuming that you're talking a "normal" type of system and we're talking basic system testing, the metric I'd go with is that 33% of all effort should be test effort. In theory this gives you a 1 to 2 ratio but assume that your developers are unit testing perhaps stretch it to 1 to 3. Certainly I'm currently running with 1 to 4 and that's not enough (I'll be rectifying it as soon as the business allows me to).
But you also want to consider how mature your software development processes are and what your specifications are like. As well as having the right number of testers you need to give them the right information to test against - if they don't have that then they're not going to be able to do a job and it's arguably wasted money.
One other option to throw in there - it sounds like the product has been under development for a while. As well as a core of regular testers, you might want to bulk up for an initial major test using short term contractors. If you're only just bringing test resource in you don't want to start them with a major backlog.