Place your connection string in the .config file for your application and retrieve it from there.
That file can be edited in each installation so that you do not require a recompile for each install.
Check out these links to help in understanding how to use these configuration files.
Storing and Retrieving Connection Strings[
^]
When deploying the application on a client machine, the .config file that is deployed with the application is a plain text file that can be opened in Notepad and simply updated with the appropriate connection string required.