From the Super Mario Wiki
Indiana Joe is the self-proclaimed greatest treasure hunter in Jungle Land. He is a spoof of Indiana Jones, although he is slightly overweight and strangely, appear to have no facial features except for his eyes, only briefly visible. However, he is quite resourceful, and knows the location of every hidden treasure in Jungle Land, as well as every shortcut leading to them.
Indiana Joe's past remains unclear, but he wound up becoming the greatest treasure hunter in Jungle Land at some point in his life, learning of all their locations in the process. It is suggested that Indiana Joe had a bad experience with Koopas at some point earlier in his life, due to his display of Koopaphobia. This may be akin to Indiana Jones' fear of snakes.
When Mario, Luigi, Toad and Princess Toadstool's hot air balloon crashed into a quicksand pond in Jungle Land, Indiana Joe rescued them with his whip and introduced himself. He then agreed to help Princess Toadstool find the Lost Mushroom, and led them to the Temple of Koopa, which he refused to enter due to his Koopaphobia.
While the others went inside the Temple of Koopa, he was inevitably scared off by Kolonel von Koop, who had followed Indiana Joe to the Temple of Koopa with the Koopa Pack. This cowardice allowed for Kolonel von Koop to seize the Lost Mushroom from Mario once he made it out of the Temple of Koopa.
Unable to continue his career as a treasure hunter due to his Koopaphobia, Indiana Joe started his own babysitting service at his home, and soon found himself looking after a trio of bratty Mushroom children. Mario's group arrived and asked for Indiana Joe to help them reclaim the Lost Mushroom back from Kolonel von Koop. Unable to take another minute of looking after the kids, Indiana Joe agreed to go along, stating that while Kolonel von Koop may be scary, he's nothing compared to babysitting three brats.
Upon finding the Koopa Pack, Indiana Joe worked with Mario to help him get the Lost Mushroom back from Kolonel von Koop by fighting him with his whip. After Kolonel von Koop retreated, Indiana Joe went home to remove his babysitting service sign, and thanked Mario's group for helping him get over his Koopaphobia. However, Mario used the Lost Mushroom to fill Indiana Joe's house with pasta, something that Indiana Joe didn't seem to fond of.