I am creating a website which until now is pure PHP. I was thinking
that since very few people do not have JavaScript enabled (which I
wonder why!) maybe I should create my website as a fully PHP site
without any AJAX. Am I thinking wrong?
I would say most people do have JavaScript enabled. Probably 2% at most have it disabled according to this post from 2012.
Just to be sure, if I implement some AJAX would it increase the risk
of my site getting breached?
Yes, technically it does. More code = more chance of bugs, and security bugs will be a subset of these.
You will also be increasing the attack surface of your application, as you will be generally be implementing more server-side handlers for user actions to be executed asynchronously.
There is also more chance of client side bugs being prevalent such as XSS (particularly more chance of DOM based XSS bugs sneaking in there).
Should I be even worried about this and start using AJAX?
You should be "rightfully concerned" rather than worried. The increased risk is considered acceptable by most sites, even high security systems such as banking applications. If implemented correctly it is possible for your site to remain "just as secure" as it was without AJAX.
As with anything web-based, pay particular attention to the OWASP Top 10 and follow secure coding practices to minimise the risks. It is always advisable to get a security assessment carried out by an external company to pickup anything you've missed, although this can be expensive.