If it is on a portal, you have to adjust the area permissions to grant
Readers (or whichever site group) the permission to "Add Items". If it
is on a WSS site, then do the same under "Change Permissions for this
survey" on the Modify Settings and Columns page.
Note on portal, changing the Add Items rights for Readers will allow
them to add a new item to any web part located on that area. Consider
storing the survey on an area of its own if this would be a problem.
Surveys by default do not allow response submissions unless the user has
Add Items rights. It is treated like any other list or library.