Connecticut vital record can get expensive. Certified vital records are $20 a copy. If you don't know where the event took place, you will have order a certified copy from the Connecticut State Vital Records Office (this only works for an event after 1897). If you do, you might be able to save yourself some money. Here are some things to consider.
1) Do you really need a copy of the record?
I know, I know. I'm committing the ultimate genealogy sin here. But consider looking at a transcription, if you don't need the original record to prove your case. FamilySearch.org offers indexes and transcriptions of Connecticut vital records. They're not perfect - vital record transcriptions often miss a few details available on the actual document - but they can help you fill in some blanks. Consider looking at the death index online, and then finding an obituary and gravestone image to help you build your case.
2)Is it an older record? The Family History Library microfilmed many records in the 1980s. Make sure you check the FamilySearch.org catalog. You can order microfilm online and have it shipped to your local Family History Center. If you need to do a lot of work in one town, this is the most economical option.