There are a few critical details missing.
- What OS were you using?
Many platforms provide or require code signing, but you don't tell us what platform you're using.
OSX will by default only run signed binaries. Windows displays the CN of the signing certificate if present (or something like that--I can't remember the precise behavior). iOS will only ever run signed binaries from the App Store unless jailbroken. Android is similar, except you can disable this. Many Linux package managers (like apt) use PGP signing to enforce signed package installation.
So if you're using such a system, you're probably safe.
- What transport did you use to download the software? You seem to imply HTTP, but this is a somewhat critical detail.
VPN'ing to some machine in a shared datacenter, as suggested in a prior answer, seems like a poor way to mitigate your exposure to an untrusted wifi network--how do you determine that you trust the (still unencrypted) connection between the datacenter machine and the software repository? The only way you can really establish trust in what you download is code signing or fully end-to-end transport layer encryption.
On the other hand, reinstalling is probably a waste of time. The odds of someone serving backdoored software on any given wifi hotspot are quite small. I wouldn't lose sleep over it.