Baskinta lies at the foot of Mount Sannine, in the Matn District - Mount Lebanon Governorate - east of Beirut. Wikipedia has alot of info/maps that will help.
I have spent the last few years trying to trace my ancestors in Lebanon - it is a slow, difficult task. You will see my name a number of times on the message board.
Firstly, Lebanon was under the control of the Ottoman Empire up until around 1918 - Lebanon, prior to this date was commonly referred to as "Syria". You might like to try the churches in Baskinta, but you might find that their records only start in the early 1900's. It appears under Ottoman control that records may have been retained centrally in "Syria". The church, however, may be able to give you some guidance as where to look. Do you know what village your ancestors came from? and where did they migrate too?
You might also try making contact with libraries/registry offices in the Matn/Mt Lebanon District/Governorate.
You will need to make sure of the "true" family name, as most changed when they arrived in foreign countries. This is important or you will struggle. It will help also if you have someone who can "communicate" in arabic to help with your enquiries.