This article is from the FAQ, by with numerous contributions by others.
For H-1B, a Bachelor's degree in a narrowly defined subject area is
sufficient. Ph.D. has NO additional significance.
For a labor cert., chances of success depend upon how many people can
qualify for the job. If an ad is placed for a Bachelors degree,
chances are a lot more people will apply than would for a job that
require a Ph.D. degree. Please remember, it is not advisable to tailor
a job to suit your own needs. If the employer asks for restrictive
requirements for the job, the department of labor will object to it.
Also, the higher the qualifications, the bigger the salary must be.
It is NOT true that ONLY advanced degrees have a good chance of success.
There are a lot of variables that need to be considered.