"To have travelled in their youth is a source of pleasurable reminiscence for elderly persons."

"To have travelled" is correct because it expresses that the action of travelling occurred in the past and this is now a source of fond memories for elderly persons. 

"To be travelling" refers to the continuous action of travelling. "To travel" refers to the act of travelling but does not express the fact that the action occurred in the past. "By travelling" suggests a means or method and not an action completed in the past.