Earlier it was working fine for these users. The users were having contributor permission level. Upon further investigation it was found that from the contributor permission level two permissions was removed accidentally. Here are those permission:
So if we remove Use Remote Interface and Use Client Integration Feature Permissions from the specific permission level the users can not use the SharePoint integration with access and excel which are associated with that specific permission level. Once these permissions are given the users were able to open the SharePoint list in Access.
Lets see what is the effect of these two permission level
These are the actions which are available when these two permission are given in a SharePoint List:
Once these permissions are removed only following options will be shown to the users in Actions menu in the SharePoint list:
Also various option on document library such as upload multiple document and Open with Windows Explorer are not available if we remove these permissions.
These are the options available when these permissions are there:
Upon removing these permission:
You can reach on any specific permission level by this path:
People and Groups -- Site Permission -- Settings -- Permission Level
Farm Administrator can also remove these permission for the entire web applications from the Central Administration. To do so go to Application Management -- Application Security -- User Permissions for Web Applications and un-check these two permission. In that case no user can access these feature on the Web Application.