It's clear from your question that you are assuming "the prize" to be salvation, and that you believe that this verse speaks of our works earning our salvation, or at least our bad works possibly leading to the loss of our salvation.
However, to most reformed theologians, this is not the case. A deeper study of the original Greek, as well as application of sound principles of interpretation clears this up.
First, the principles of Biblical interpretation include the fact that Scripture cannot contradict itself. Paul is on record all throughout the New Testament as stating that salvation is by faith through grace, apart from any works or worthiness of man. It would make no sense for him to be saying so here, so we need to dig deeper.
Taking a look at the various commentaries on this verse, most are in agreement that the prize is not salvation. For example:
From Scofield Reference Notes
Gr. adokimos, "disapproved." Dokimos, without the private a, is
translated "approved" in Rom 14:18 16:10 1Cor 11:19 2Cor 10:18 2Tim
2:15 Jas 1:12, by the word "tried." The prefix simply changes the word
to a negative, i.e. not approved, or, disapproved. The apostle is
writing of service, not of salvation. He is not expressing fear that
he may fail of salvation but of his crown. See "Rewards" Dan 12:3 1Cor
Another view from Gill's Exposition of the Entire Bible hints that the prize is simply faithfulness and remaining a faithful servant so that he can achieve the ultimate prize of expanding Christ's kingdom - avoiding becoming a "useless preacher".
I myself should be a castaway, or rejected, or disapproved of; that is,
by men: the apostle's concern is, lest he should do anything that
might bring a reproach on the Gospel; lest some corruption of his
nature or other should break out, and thereby his ministry be justly
blamed, and be brought under contempt; and so he be rejected and
disapproved of by men, and become useless as a preacher: not that he
feared he should become a reprobate, as the word is opposed to an
elect person; or that he should be a castaway eternally, or be
everlastingly damned; for he knew in whom he had believed, and was
persuaded of his interest in the love of God, and that he was a chosen
vessel of salvation, that could not be eternally lost: though
supposing that this is his sense, and these his fears and concern, it
follows not as neither that he was, so neither that he could be a lost
and damned person: the fears of the saints, their godly jealousies of
themselves, and pious care that they be not lost, are not at all
inconsistent with the firmness of their election, their security in
Christ, and the impossibility of their final and total falling away;
but on the contrary are overruled, and made use of by the Spirit of
God, for their final perseverance in grace and holiness.
Calvin put the same thought this way:
Accordingly, I strive to conduct myself in such a manner, that my
character and conduct may not be inconsistent with my doctrine, and
that thus I may not, with great disgrace to myself, and a grievous
occasion of offense to my brethren, neglect those things which I
require from others.
Finally, if you're interested, there's no shortage of commentaries from the great thinkers here: http://www.preceptaustin.org/1_corinthians_927_commentary.htm