Many people hire workers to build their Sukka for them in preparation for Sukkot, which is perfectly acceptable. The question arises, however, as to whether a Sukka is valid if it was built entirely by a non-Jewish worker. The primary component of the Sukka is the Sechach (covering), and one might thus assume that this part of the Sukka must be put in place specifically by a Jew, who is obligated in the Misva of Sukka.
The Gemara, however, indicates otherwise. In Masechet Sukka, the Gemara establishes that as long as the Sechach was placed for the purpose of providing shade, the Sukka is valid, regardless of who placed it. And thus even if a non-Jew put on the Sechach, and he has no knowledge whatsoever about the Misva of Sukka, the Sechach is valid, since it can be assumed that the worker understood that the covering is intended to provide shade. The second condition is that the Sechach is placed within thirty days of Sukkot, such that it is obvious that it is placed for the Misva.
Although Rav Haim Palachi (Turkey, 1788-1869) maintained that one should not have a non-Jew construct the Sukka, Hacham Ovadia Yosef, in Hazon Ovadia (p. 56), disputes this view and shows that it is perfectly acceptable for a Sukka to be built by a non-Jew. This was also the ruling of the Rif, the Rambam and the Helkat Yoab, and this is the opinion accepted by Hacham Ovadia both in Hazon Ovadia and in Yalkut Yosef – Sukkot (p. 110; listen to audio recording for precise citation).
It should be noted that in general, it is preferable to perform Misvot personally rather than discharge one’s responsibilities by having others do the work for him. This point was made by numerous Poskim and by the Arizal (Rav Yishak Luria of Safed, 1534-1572). Hence, it is certainly preferable for one to personally involve himself in the building of a Sukka. Nevertheless, if one has a non-Jew build his Sukka, the Sukka is perfectly valid for the Misva.
Summary: Although it is preferable to personally involve oneself in the construction of the Sukka, as is the case regarding all Misvot, nevertheless, a Sukka built by a non-Jew is perfectly valid, even if it was built entirely by the non-Jew.