I've been trying to figure out exactly when U.S. Customs started searching the belongings of travelers. In researching the matter Palindrome and I found this short piece which was published by the New York Times in 1879. Not only does it describe an inspection process similar to the modern one, but it mocks it quite mercilessly.

There is no doubt that the real object which our Government had in view in establishing the existing rules in regard to passengers' baggage is not the collection of revenue but the education of the traveling public in the elegant and useful accomplishment of perjury.

This essay is a useful datapoint, but I'll have to push further back in time for my answer.

(When you open the PDF, scroll down - the article starts in the left column.)

Here is a reference from 1864.

There was but one obstacle that we foresaw would interfere with our immediate departure from New York, and that was the examination of the passengers' baggage, which we anticipated would be diligently overhauled by the officers. Doubtless through the heavy demands on the United States Government for the continued support of the expensive war now being waged upon this continent, a strict and rigid system of searching emigrants' luggage for contraband articles, is enforced; the officers charge for almost everything besides what individuals are clothed with.

A prewar reference to customs inspections of travelers.

Emigrants who prefer going into Canada by way of New York will receive advice and direction by applying to the British Consul at New York (James Buchanan, Esq.) Formerly this gentleman could procure for emigrants who were positively determined to settle in the Canadas, permission to land their baggage and effects free of custom-house duty; but in a letter dated 16th March, 1835, he says-
"In consequence of a change in the truly liberal course heretofore adopted at this port, in permitting, without unpacking or payment of duty, of the personal baggage, household, and farming utensils of emigrants landing here to pass in transit through this state to his Majesty's provinces, upon evidence being furnished of the fact, and that such packages alone contained articles of the foregoing description, I deem it my duty to make known that all articles arriving at this port accompanying emigrants in transit to Canada, will be subject to the same inspection as if to remain in the United States, and pay the duties to which the same are subjected. I think it proper to mention that all articles suited to new settlers are to be had in Canada on better terms than they can be brought out and such as are adapted to the country."

A GALLANT DEED. - Article Preview - The New York Times