Cleaning an Antique Christening Dress

It depends on the fabric. I agree about contacting a local museum. You can also check on the internet on how to clean antique clothing. You need to be very careful with it.
I used to collect antique clothing and there are books on the subject.

Since this is delicate and sentimental please don't try to clean it yourself. Find a good dry cleaner that clean with "archival" quality products. If you have a local art museum the curator might be able to suggest one.