Here's my big worries about a potential reboot:
1) That Disney will decide to set the reboot in the present.
Look at the new Zorro reboot; it's being reinvented as a post-apocalyptic story set in the future. I worry Disney will decide to jettison the 1920s-1950s range of time and just go straight to today's world, reinventing Indy as played by Chris Pratt as a modern-day adventurer.
2) That Disney will get the 1930s-1940s wrong.
Part of the charm of the Indy films is that they look and feel generally period appropriate. The lush tones of Last Crusade recall the Technicolor of the late 1930s; KOTCS purposely went for a 50s film sheen; All of the original 4 movies are very period appropriate, both in terms of manner of other characters and the attire etc. They're as perfect as period films as they are as adventure films. I worry an Indy done by Disney won't "get" the era right, or understand the genre antecedents to Indy/which inspired it.
3) That Disney will attempt to artifically "diversify" the cast/be politically correct
While I'm all for a minority character if the character has merit and is written well, I'm not for characters being minorities just to fill some random political correctness quota. Part of the fun of Indy is that it isn't politically correct. Feminists today would HATE Willie, would pillory Marion for falling in love with Indy; would have a fit that a woman was shown to be a Nazi sympathizer; Social Justice types would have a fit that the only African characters in the cast were pirates. Not only that, but would Disney ever make a film as daring or unpolitically correct as ToD? Would they dare to show Nazis or Soviets as bad guys? That's treading on being politically incorrect.
"This is 2015, the Cold War is over - we can't show Russians in any villainous or negative way, think of the international turmoil it could stir up! China makes most of our products here at Disney - No Chinese gangsters in our Indy film! Nazis are just too un-politically correct to even touch in a movie! It has to be a family friendly adventure romp to sell toys and promote our theme parks! Nazis make people automatically think of the Holocaust - we can't have that associated with our toy line.
We can't show Indy fighting someone and having blood on him or the other guy being bloodied! If he fights, it has to be clean, no scuffs or scrapes on he or the bad guy- think of the children in the audience! We can't show a heart being ripped out or a man aging rapidly or a woman being disintegrated - Our focus groups show us that such won't go over well with our target audience!
Indy needs a token black friend or else the franchise will be deemed racist. The next Indy girl needs to be strong, empowered, and smart - smarter than Indy. She can't fall for him - Indy has to fall for her!
The artifact in the new Indy movie can't be religious or particular to any one religion, because then we'll be offending audiences of other faiths. If the Ark of the Covenant is real, what does that say to the Buddhists of the world? The Atheists? If the Holy Grail is real, what does that say to Muslims and Jews? We can't have Indy go looking for a religious artifact - that's too politically incorrect!"
4) That their Indy will be either too light or too dark, and will be shallow.
Look at their Marvel movies - fun, but very goofy. That's what I've seen of Disney doing action films as of late. The heroes are a bit on the goofy side or are extreme caricatures (Tony Stark is a barrel of snarky laughs, Captain America is a laughable goody-two-shoes stuck in an innocent past).
Or, they could take Indy and Nolanize it, going way too far into the ponderous and dark; making an Indy film akin to the Craig Bond's - dark, overly serious, overly brooding.