The reason for this "no" depends on whether we're talking about an area where public transport is highly developed (e.g. most urban areas of Western Europe and Japan) or areas in which it is underfunded and underdeveloped and is mostly seen as transport for poor people (e.g. most of the USA except NYC).
In the case of developed public transportation the answer is no, because simply adding first class would decrease capacity, which at rush hour is already at its limits. Hence, while a first class may attract a few new users, it would necessarily decrease the overall capacity of the system resulting in fewer users overall.
In the case of underdeveloped transportation, the answer is also no. The reason is that simply adding a first class to an insufficiently planned and inefficient network would still be little more than a more comfortable alternative for the less poor users to use said inefficient and insufficiently planned network through the city. In other words, the only people who would use it were those who were already doing so, be it out of principle (environmentalism), necessity (foreign students) or convenience (the bus just happens to be the perfect route to work and they can read a book on it), but who can actually afford this newly introduced first class. So, no net gain in terms of users.
If you want to increase the number of users of public transportation you don't need first class. You need more capacity and better and more efficient networks.