When Barnaby and Betty are in the lab, he tells her he's trying to develop a compound that would crystallize fingerprint oil on any surface. What he's describing became reality a few years later. In 1978 it was shown that cyanoacrylate (super glue) fuming would indeed crystallize fingerprints on porous surfaces like cloth, or even skin, making them stable enough to be dusted and collected. The process is now widely used (2021).
Barnaby often travels to Mexico because he likes the food and enjoys the fishing. Fishing is a popular hobby for many TV detectives of the era including Barnaby's good friend Frank Cannon.
Although Barnaby Jones (as well as Mannix, Cannon, Rockford and most similar private detective characters) is often involved in car chases in which his car would likely get damaged, such characters are rarely seen driving a damaged car. They must have very fast body shops and tolerant insurers. This (at about the 40 minute mark) is one of the few episodes in which Barnaby can be seen getting into a car that's been conspicuously side-swiped.