It can be very annoying having users accidently close web parts. It’s a long process to restore them.
To re-open a closed web part:

Go to Site Actions > Edit Page:
o Add a web part
o Advanced Web part gallery and options
o Select “Closed Web Parts”
o Drag-n-drop the web part back onto your page

To prevent this process from happening, try turning off the close web part button. It doesn’t remove the possibility of a user closing the web part but makes it much less likely to accidently happen!

For Each Web Part:

1.    Click on the web part menu and choose “Modify Shared Web Part” (as shown below)

2.    Expand “Advanced”

3.    Uncheck “Allow Close” (see below)

4.    Press OK