Can I update to Ruby 2.1.2 using Rails 3.2.3?

According to the Travis configuration, Rails 3.2.3 was only tested with Ruby 1.8.7, 1.9.2 and 1.9.3, so it doesn't seem to be a good idea to use a later version of Ruby (but you can always test it yourself.) However, Rails 3.2.3 has some known security vulnerabilities that have been patched in 3.2.19.

On the other hand, the configuration for Rails 3.2.19 does suggest that the developers expect it to work with Ruby 2.0.0. Your project might benefit from investing some effort to see if you can upgrade from Rails 3.2.3 to 3.2.19, and test a combination of that with Ruby 2.0 for your application, if there are new Ruby features that you need and cannot efficiently backport them. Bear in mind that the only recent answer to a similar question warns of 'weird issues' from such a combination. Also consider that the latest Rails 3.2 release notes do not mention Ruby 2.0.

The first release of rails that officially support ruby 2.0 was 3.2.13 (see the announcement on the rails blog.

I deployed several applications running 3.2.15-3.2.17 and ruby 2.0 (They've since been upgraded to rails 4) without any problems that I recall.

The recently released 3.2.22 supports ruby 2.2 (announcement)