Stay out of it and keep out of it.
Their application is their application and it has to be evaluated on its own merits. Nothing to do with you.
Don't say anything to your new employer, you have nothing to say that they don't already know anyway. Weigh in at some time in the future only if your colleague is much further long in the hiring process, you are being solicited for your opinion, and you know your colleague to be an out and out jerk. Ditto if the colleague as a pleasure to work with. In general, respect the confidentiality of your new company's hiring process.
You told your old boss who you were to work for. Nothing wrong with that in my book - I see it as a matter of (non-obligatory) professional courtesy to tell the old boss where you're going. Your colleague got wind of of it, most likely from your boss (*). I am inclined to say there's nothing wrong with that. And your colleague is applying to your new employer. Again, nothing wrong with that. You are not a party to your colleague's decision to apply nor are you likely to be a party to your new employer's decision to hire - and that's probably the way you want it.
Frankly, I don't understand what you are worried about. So far as I am concerned, your worry is much ado about nothing. If your colleague is qualified and your new employer wants them, that's 100% your new employer's decision. You have nothing to do with that decision, and your new employer has certainly every right to hire your colleague away from their present employer.
(*) Keep in mind that your suspicion that your boss talked is just a theory, and you have no proof that this is actually what happened although it is the most likely scenario - Your colleague could have learned from a sister-in-law who works at your new company for all we know. Having said that, how your colleague learned about the existence of your new employer is irrelevant to you, to your new employer or to anyone else. It's not as if your new employer is a secret society of some kind, is it?