If you are a US citizen, you can indeed cross into Canada using your birth certificate and a photo ID. For safety reasons, you may want to bring the original birth certificate, or at least a certified copy. If you are not a US (or Canadian) citizen or permanent resident, you cannot enter Canada without passport:
If you are a citizen of the United States, you do not need a passport
to enter Canada. However, you should carry proof of your citizenship,
such as a birth certificate, certificate of citizenship or
naturalization, as well as photo identification. If you are a
permanent resident of Canada or the U.S, you should bring your
Permanent Resident Card with you.
However, depending on how you're travelling, you may have difficulties re-entering the US without a passport! These are the rules for US citizens coming back:
When returning to the United States from Canada, it is very important
to note that all U.S. citizens are required to present a valid U.S.
passport to enter or re-enter the United States via air. For entry
into the United States via land and sea borders, U.S. citizens must
present either a U.S. passport, passport card, NEXUS card, Enhanced
Drivers License, or other Western Hemisphere Travel Initiative
(WHTI)-compliant document. The only exception to this requirement is
for U.S. citizens under the age of 16 (or under 19, if traveling with
a school, religious, or other youth group) who need only present a
birth certificate (original, photocopy or certified copy), Consular
Report of Birth Abroad, or naturalization certificate.
So if your passport is delayed, make sure it arrives at your place in Canada on time.