In this situation, there is really only so much that you can do. Ultimately, it is your boss’s responsibility to know better than to get drunk when meeting with an important client. If your boss does have one drink too many, his attitude will largely determine what you can do to prevent him from embarrassing himself.
The first thing that you could do to prevent this problem is to prevent it from happening in the first place. However, this is very difficult to do. In order to prevent it from happening, you would have to see that your boss has had enough to drink and somehow dissuade him from ordering or consuming another drink. This would be extremely difficult as your boss might be insulted that you think he is getting drunk or that you think that it is your place to tell him how much to drink. This could end very badly for you.
If your boss does get drunk, your best hope is to minimize the impact of his drunkenness. If he starts to talk about things that are off-topic or which might get him in trouble, try to steer the conversation back to safer/more appropriate grounds. If you can’t get him to stop embarrassing himself, you can try to give him a cover story. Say something like “Boss, I forgot I was supposed to remind you not to have a drink tonight because of the medicine you’re taking.” Perhaps then you can pass it off to the client as an unfortunate reaction to mixing prescription medication with an innocuous amount of alcohol. You can also pass it off as your fault and not that of your boss. The problem with both of these is that your boss will need to cooperate with you which might be hard given that he is your boss and he is drunk.
Again, however, there is only so much you can do in this situation. This is something that your boss needs to be smart enough to avoid.
If faced with this problem in that context, you are indeed placed in a difficult situation since it may be insulting for you to be the judge of how many drinks your boss should consume as he should be the one to determine this knowing his own alcoholic tolerance level. However, being in the position of assistant, it is clearly your responsibility and duty to prevent your boss from embarrassing himself because of consuming excessive alcohol. One way may be to politely excuse yourselves or suggest that it seems everyone has had enough alcohol tonight and may want to instead drink some nice hot tea to end the night.
Well personally I would stop them from drinking too much. Really the best way to stop the embarrassment from happening is to make sure that your boss stays sober. Watch what your boss does to make sure he/she stays professional and remind him/her, in a polite way, that they needs to watch how much they is drinking.
If he/she does happen to drink to much, make an excuse as a reason to leave or maybe just talk with business professionals for them, you are the assistant after all and it wouldn't be uncommon.
In this setting, there are only a few things you could do since you are not in the position to talk to your boss about his actions, his job responsibilities and his ability to hold his liquor -- you are his assistant. In a very formal event such as a dinner party, he is expected to be at his best and nothing less. However, if he gets a bit tipsy or drunk even, as the assistant, you could not save him from himself. It is his responsibility to keep himself together. The best thing you could do is be by his side all night and make sure he still says the right things, and he does not insult any of your business partners.
The best solution would still be prevention. If you have been his assistant for a long time, you should know just how much he can drink before he gets tipsy. So make sure he never drinks any more than that and know the agenda of your business so maybe, if all else fails, you could try talking on his behalf. But doing so might not look very good especially for your boss.
Also, when he is sober, maybe you could actually ask him politely not to get drunk because it affects the business.