Browser fingerprinting consists into collecting attributes from a web browser. Hundreds of attributes have been discovered through the years. Each one of them provides a way to distinguish browsers, but also comes with a usability cost (e.g., additional collection time). In this work, we propose FPSelect, an attribute selection framework allowing verifiers to tune their browser fingerprinting probes for web authentication. We formalize the problem as searching for the attribute set that satisfies a security requirement and minimizes the usability cost. The security is measured as the proportion of impersonated users given a fingerprinting probe, a user population, and an attacker that knows the exact fingerprint distribution among the user population. The usability is quantified by the collection time of browser fingerprints, their size, and their instability. We compare our framework with common baselines, based on a real-life fingerprint dataset, and find out that in our experimental settings, our framework selects attribute sets of lower usability cost. Compared to the baselines, the attribute sets found by FPSelect generate fingerprints that are up to 97 times smaller, are collected up to 3,361 times faster, and with up to 7.2 times less changing attributes between two observations, on average.