Altaïr's fortress and home base in Assassin's Creed is Masyaf. The mountain settlement serves as the headquarters of the Assassin Brotherhood during the game's Third Crusade storyline.
Masyaf is based on a real fortress in modern-day Syria. Historically, the castle was associated with the Nizari Ismailis, a medieval religious community whose members were often labeled “Assassins” in Western sources. The game fictionalizes this history and connects the fortress to Altaïr and his mentor, Al Mualim.
Altaïr returns to Masyaf after missions and receives assignments there. The fortress also functions as a symbol of the Brotherhood's organization, discipline, and secrecy. Its elevated location contrasts with the major cities where Altaïr carries out investigations and assassinations.
Acre, Damascus, and Jerusalem are important mission cities, but they are not Altaïr's permanent headquarters. Masyaf is therefore the specific answer when the question asks for his home base rather than a city visited during the campaign.