Specifically, in the early 20th century and before. And I don't mean immigrants/etc who traveled with all the things they owned not, probably, the super rich, who would just have servants handle their money. I mean a wide swath of the middle-lower upper class. If they took a steamship across the Atlantic or went on safari in Africa or adventures in India, how did they pay for everything? I would presume they would have to bring all their money with them, no? If so, weren't obvious European tourists in Africa/India rife for robbery? Was there a way to send money? And going back ever farther, what about pilgrims or travelers hundreds of years ago or in Roman times? They'd just have to bring money for the duration of the trip, no? Did this create problems and/or opportunities?
It's almost not a question of history as the answer for the early 20th century is the same as for the late 20th century: Travellers' Cheques. Travellers' Cheques offered greater security than cash for large sums and we're method of choice for taking larger sums of money abroad as they offered greater security as they were signed beforehand and then signed again in the presence of the post office or bank clerk who was cashing it while you also presented ID. Prior to the rise of credit cards in the 60s, American Express was most famous, internationally at least, for being one if not the, dominant suppliers of travellers cheques.