Using public internet will effect your security. This is because all traffic that is traveling on the network unencrypted can be sniffed by anyone else who has access to the network.
So long as you log in to any online accounts through a secure https connection your passwords will at least not be visible to an attacker in plain text.
Using Tor/Tails is also an effective way to encrypt all of your traffic on the LAN because traffic is encrypted as it travels through the tor network from your PC to the tor exit node as shown in "how tor works 2" in the tor overview https://www.torproject.org/about/overview.html.en however it is still best if the websites you visit are https as this way you have point to point encryption.
To give some alternatives you could use a VPN (Virtual Private Network) which is another way to encrypt your traffic on the LAN but usually costs money and doesn't provide the same anonymity as tor. Or someone with access to a remote server could tunnel through ssh to create their own proxy server. There are videos explaining how to do this on youtube, like this one https://www.youtube.com/watch?v=__VASPeU82Q
The other thing you need to watch out for if you are in a public place when you connect to the public internet is visual collection of your passwords. If someone or a camera can see your keyboard as you're typing in your login details then they can remember or write down your passwords, or re-watch the camera feeds at their leisure to gain everything they need to access your accounts.