By default SharePoint will disable session, if we have to use session in web parts, then how should we do so? If I don't use session, then is there any alternative solutions ?
Here are some considerations:
You could enable the session. There are several blogs or forum posts about that (like https://stackoverflow.com/questions/2237989/enabling-session-state-in-sharepoint-2010).
But enabling/using the session may have an impact on the performance, especially if you have several front end servers.
Maybe try to use the ViewState/Cookies instead of the session state.