How should we judge the trustworthiness of a given Bible translation? Given that both the humans who actually penned the Bible and the later humans who translated it all had free will, the issue of Biblical inerrancy is a tricky one.

See also: By what mechanism could the Bible be inerrant?

It is usually claimed that those transcribing and translating the text were working under the guidance of the Holy Spirit. While this may be true of some, it is also clear that not every translation project has even approached the work in good faith, much less under supernatural guidance. Some 'translations' are clearly contradictory with others, and others show obvious signs of a specific agenda being brought to the text. Others specifically state having a political bias going into the project (e.g. the Conservative Bible Project). Of course nearly every group or individual that produces a translation then goes on to claim that theirs is the "true" rendition of the text or meaning.

Since clearly not everyone that works on the Bible is guided by the Holy Spirit, how are we to know which translations are reliable? Are modern and ancient mainstream translations considered to be free from political agendas and personal bias? If so, how do we know?
Cliche Christian answer: pray about it. I'm guessing that isn't the answer you're looking for though, so I will say "research."

For example, looking at Wikipedia you can see the New International Version "translation took ten years and involved a team of up to 100 scholars from the USA, Canada, the United Kingdom, Australia, New Zealand, and South Africa. The range of those participating included many different denominations such as Anglicans, Assemblies of God, Baptist, Christian Reformed, Lutheran and Presbyterian." Also, "Recent archaeological and linguistic discoveries helped in understanding traditionally difficult passages to translate."

Having a large team of international, inter-denominational scholars helps quite a bit to remove bias and helps to ensure an accurate translation. Then, using different discoveries gives further insight.

You can easily do this research (and more in depth research) for the other translations as well. I only used NIV as an example because it is what I see quoted most often. With so many scholars using so many different translations, it is usually pretty easy to find out when a translation is inaccurate.

If worried about bias, consider purchasing a parallel Bible so you can always compare two or three translations