My daughter just turned 19 today. She and her boyfriend went to Canada, where the legal drinking age in some provinces is 19. She just asked me "Could I get in trouble for being intoxicated, if we get pulled over, after we cross the border back into the US"? Mind you she wasn't driving. I didn't have an answer for her. I actually thought it was a valid question.
It depends on which US state she is returning to and yes, she may get in trouble. There are US states at the Canadian border not only prohibiting sales, purchase, possession or consumption of alcohol to or by underage persons, but also strictly prohibiting the state of actually being intoxicated, e.g. Idaho, Michigan and New Hampshire. Other states have statutes with similar effect, e.g. prohibiting recent consumption (North Dakota) or exhibiting the effects of having consumed alcohol (Washington).
You can find further details regarding underage drinking in state laws in NIAAA's Alcohol Policy Information System.
Edit: I thought it would be enough to link to a government organization like NIAAA, where the applicable laws are both quoted or summarized and also referenced. Since Lemuel Gulliver still disputes the existence of such laws in Idaho and North Dakota in his answer, here are the relevant quotes.
Idaho Statute 23-949:
It is unlawful for any person under the age of twenty-one (21) years to ... possess ... beer, wine or other alcoholic liquor. For purposes of this section, a person shall also be deemed to "possess" alcohol that has been consumed by the person, without regard to the place of consumption.
North Dakota Century Code 5-01-08(1):
An individual under twenty-one years of age may not ... consume or have recently consumed ... [or] be under the influence of ... an alcoholic beverage.
I could not find the statutes which Tor-Einar Jarnbjo is referring to in his answer. As far as I know it is only illegal in Idaho for a minor to drive under the influence of alcohol. For North Dakota I reviewed the code sections 5-02-06 and 5-01-08(1) and did not find any prohibition for a minor to merely be intoxicated with alcohol.
In general, all states outlaw "public drunkeness" whether the person is a minor or not, so if you fall afoul of that it does not matter your age.
A border guard will not enforce state laws, they are federal agents.
In general, simply being drunk in a private place is not illegal. If you want a specific answer you must specify the STATE in which you will be and the situation. Are we in a bus? in a car? in a plane? walking on the road? You need to specify the circumstances and the reason why you would be stopped. For example, if a car is stopped for speeding, the police have no right to question the passengers in the vehicle or test them for alcohol regardless of whether they are minors or not.
There is a difference between being arrested and being convicted of a crime. If a US police officer observed a drunken 19 year old, they might choose to arrest that person, suspecting them of having drunk under age. If the drunken teen blurted out "no, I drank in CANADA! It's totally fine!" I suspect the officer would pay little or no attention to it, though they might make a note of in in the arrest notes.
It's possible that later, in the police station, this protestation would have the desired effect and the teen would be released. It might not, and could be raised later at a trial, where evidence of having been in Canada would be admissible. You might even end up with the case dismissed.
But is that really a good outcome? Getting arrested is not very pleasant. Only spending a little time in the lockup, or ending up not getting a criminal record, isn't much of a consolation prize. "Getting in trouble" is not a simple as not doing anything illegal. You can have an unpleasant interaction with authority even if there's no reason to suspect you've done anything wrong. Providing evidence that many officers would see as clear evidence of a crime is foolhardy.