When clicking on the Drive icon in Google Classroom you can search folders but you are not able to natively attach a folder.
Attaching a Folder
To attach a folder in Google Classroom you will need to start from Google Drive (http://drive.google.com). Locate the folder you wish to share and click on it one time. This will reveal a toolbar of options along the top.
After single clicking on the folder in Google Drive, click on the share icon. This is an image of a person with a plus sign.
This brings up the share dialogue box. Enter in the email addresses of the students in the class to explicitly share the folder with the students. Alternatively, click on the "Advanced" link in the bottom right-hand corner.
On a Google Apps domain you have 6 choices for sharing. You can share with anyone in the world or just to people on the domain. If you are looking to share the folder in Google Classroom, sharing on the domain makes the most sense. All of the students in your Google Classroom class are logged in on the domain.
Can Edit or Can View
Under the link sharing options is a drop down list to choose to share the folder as "Can view" or "Can edit." When I want to share a folder of student work so other students can do peer evaluation, I will choose "Can edit." If I am just sharing resources I leave the default "Can view."
The sharing link is located in several places. From Google Drive there is a link icon in the toolbar that can be copied. From the sharing screen the link is provided in a couple of places. Copy the link to the folder.
Link in Classroom
For students to submit an entire folder in Google Drive they would also need to check the sharing permissions on the folder. The student will need to ensure the folder is shared with the teacher. The link to the folder can be submitted by the student clicking on "Add" and choosing the link option to attach the link to an assignment.