"Caucasian" is an obsolete ethnological term, much like how "White" and "Black" are obsolete ethnological terms.
As ethnicity is tied with linguistics and culture, the more proper terms are derived from the language families. Indo-European, Semitic, Altaic, Nilo-Saharan, etc.
Jews, along with Arabs, Egyptians, and various Mesopotamian tribes are Semitic and have similar physical appearance, languages, and cultural customs. Jesus, as a Jew, likely had a similar appearance to modern Palestinian Jews and Arabs.