How do you set directory permissions in NSIS?

Use the AccessControl plugin and then add this to the script, where the "logs" directory is in the install directory.

AccessControl::GrantOnFile "$INSTDIR\logs" "(BU)" "FullAccess"

That gives full access to the folder for all users.

AccessControl::GrantOnFile "<folder>" "(BU)" "FullAccess" didn't work for me on a Windows Server 2008 machine. Instead I had to use this one:

AccessControl::GrantOnFile "<folder>" "(S-1-5-32-545)" "FullAccess"

S-1-5-32-545 is equivalent to "Users" according to Microsoft Support: Well-known security identifiers in Windows operating systems.

Instead of changing the permissions on directories under Program Files, why not put the logs in a location that is writeable by all users.

See the 4.9.7.7 SetShellVarContext section in your NSIS documentation. You can use it with $APPDATA to get the application data folder that is writeable for all users.