My answer covers the meaning of this expression, including its social connotations. How to express the in English is off-topic on this site, and there isn’t enough context in the question anyway.

“Please have him receive my cordial handshake” is a correct literal translation. In other words, “please shake his hand on my behalf”. The cultural implication is “assure him that I am his friend”.

Depending on the exact culture involved (“in U.S.” and “in Europe” are not precise enough: where are the people from? Are they following their own customs or the customs of the places they’re visiting?), a handshake may happen when you first meet someone (as in contemporary US), when you first encounter the person for the day (as in contemporary France), or some other variation. If you want to understand the subtext at this level of detail, you’ll need to do more research on the specific situation.

In modern French, one does not send virtual handshakes through the mail. But this was done at the time. For example, in Les Usages du siècle by “Une Parisienne”, published in 1895, we find “Cordiale poignée de main” as one of many common closings for letters.

Here are a few examples of published letters using this expression:

• 1852 letter from a military officer to a friend
• 1869 dedication of a scientific article to a colleague
• 1892 letter to a fellow socialist militant

As you can see from these examples, there is no particular implication in terms of social class.