To answer the question effectively, one needs to analyse both the immediate and general context within which the word is used.
The common definition for 'antique' is 'old' or 'not modern'. It can also mean 'dating from a period long ago'. This means that the word refers to something that is not current or recent. In the poem, the word is used in the following line:
I met a Traveler from an antique land,
The word in this context denotes an 'old' land, that is, a country or area that has a history which goes deep into the past - an ancient history. We know such a country possesses such a history because records of its past have been discovered and studied by eminent archaeologists, palaeontologists and historians, all experts in their respective fields. There also still exists physical evidence of its history, such as statues, structures and so forth, such as, for example, the pyramids in Egypt and even the Inca temples in South America. Furthermore, carbon-dating also provides a more or less accurate frame of reference. In this instance, a country without such historical depth, cannot be defined as 'antique'.
What the first line says, in immediate context, is that the speaker had met someone who was either a resident from such a land, visiting the place where the speaker and he met, or that the person is a tourist who had just recently visited such an old place. This definition is therefore neither positive or negative, but neutral. One cannot ascribe any specific mood or tone to the specific word since the speaker is stating a simple fact.
However, it is what the traveller divulges that has a negative connotation, for he informs the speaker that he had seen a huge, broken stone statue in this old land, depicting the face of a seemingly uncaring, brutal and arrogant leader. The statue was 'trunkless' and only its face, legs and a pedestal carrying a legend had survived. The wording on the pedestal confirmed the identity as well as the arrogance of its previous owner.
In the greater context of the poem then, it could be said that 'antique' has a negative connotation, for two reasons. Firstly, it creates the impression that ancient rulers were abusive, materialistic and overpoweringly domineering or tyrannical as well as arrogant. They were overwhelmed by their own importance and believed they were demigods.
Secondly, the word has a negative connotation in the sense that the haughty leader mentioned in the poem failed miserably in his attempt to permanently assert his authority, for nothing remains of his power, except broken pieces of stone, in a vast, lonely desert. This dictator has therefore remained in the past, he is 'antique' and bears no relevance or power any more. For him then, the descriptor 'antique' is negative.
Alternatively, 'antique' in this wider context also has a positive meaning. Because this dictator has disappeared into the annals of time, the reader and the speaker should be glad and grateful that this leader with a 'sneering cold command' has remained part of ancient history and does not pose a threat any longer. Good riddance.