TeamViewer Security -How secure is it for simple remote support?

There's a couple of differences between using a 3rd party supplier (such as teamviewer security) and a direct remote control solution (eg, VNC)

TeamViewer Security has advantages in that it doesn't require ports to be opened on the firewall for inbound connections, which removes a potential point of attack. For example, if you have something like VNC listening (and it isn't possible to restrict source IP addresses for connections) then if there is a security vulnerability in VNC, or a weak password is used, then there is a risk that an attacker could use this mechanism to attack your customer.

However, there is a trade-off for this, which is that you're providing a level of trust to the people who create and run the service (in this case Teamviewer). If their product or servers are compromised, then it's possible that an attacker would be able to use that to attack anyone using the service. One thing to consider is that if you're a paying customer of the service, you may have some contractual come-back if they're hacked (although that's very likely to depend on the service in question and a whole load of other factors)

Like everything in security, it's a trade-off. If you have a decently secure remote control product and manage and control it well then I'd be inclined to say that that's likely to be a more secure option than relying on a 3rd party of any kind. That said if the claims on TeamViewers website are accurate it seems likely that they're paying a fair degree of attention to security, and also you could consider that if someone hacks TeamViewer (who have a pretty large number of customers) what's the chance that they'll attack you  TeamViewer Security - Best Practices

Exit TeamViewer Security, and Run It Only When You Need It

Our first suggestion is both an immediate action you need to take and a general suggestion for future use. First, compromises are often a result of poor security practices, we’re going to do one thing right away: shut TeamViewer temporarily off and update it, and, while the application is turned off, we’re going to update the security on your TeamViewer account through the company’s webpage. (More on this in the next section.) As a general future consideration, only run the TeamViewer application when you need it. That way, even when there is a vulnerability in the program (like the one just discovered and patched), you won’t be in nearly as much danger. An application that isn’t running can’t cause any trouble for you. While we understand that some people keep TeamViewer on 24/7 as part of their workflow, and if you absolutely have to, fine. But if you only use it occasionally in your home, or you’re one of the people who only turns it on to occasionally troubleshoot a relative’s computer, then don’t leave it running all day, every day. This is the single best way to avoid giving someone access to your machine.