Be careful with absolutes here. A lot depends on what you are doing & where the inefficiencies may lie.
In a Web Page where each user has a separate security context you may have no choice but to establish a new SQL connection with new security credentials with each page hit. Clearly nicer if you can use a pool of SQL connections with a shared security context & let the Web page filter the results but perhaps you can't.
In early versions of SQL Server ie (v6.5 or less) the Login Authentication was done by SQL Server. Also SQL was severely constrained by connection memory & the number of active connections it could handle. So it was a great idea to drop your connection when not in use.
Post v6.5, most people use Windows Authentication to login to SQL. This causes a lot of network calls between servers & some latency. Kerberos Security is even more chatty, Thus establishing a SQL connection is expensive. For that reason you need to find a balance between Holding a connection open for the life of your WinForms application vs Opening & closing it within each method call.
As a rough guide, if you think your app is going to want to talk to SQL in the next, say 30 secs. Keep the established connection open. If they've minimised your app, not touched it within a timeout period, or you've got all the data in RAM & they are unlikely to need anything more from the SQL system. Close the connection.
Consider creating a Class with a System Timer to hold the connection. Your class will always provide a valid connection, but perhaps the class will choose to drop it & free the connection load on SQL when appropriate.
Unless you are also writing Server based code, a small amount of memory inefficiency might not even be noticed. But 2-10,000 clients all poorly using your Security & Data Servers is likely to bring your Data Centre to its knees.