Personally, I believe every citizen should get and then maintain a passport. There are great travel options around every corner and if you already have a passport, you'll be ready to fly or sail or walk at a moment's notice.
From the camino standpoint. No matter where you're starting your journey, you'll need identification. And, if you're not a resident of Spain, you'll need a passport.
Passports are relatively easy to acquire in the US. You fill out a form, have a few small pictures taken, and you either mail the form into the US Passport office. You can also take it to a USPS office and submit it, but there's an extra fee.
Passports are issued by the US Department of State. They have a user friendly website here: https://travel.state.gov/content/travel/en/passports.html/ with step-by-step instructions and easy to follow tabs for new passports, renewals, and FAQ.
The website says it takes 6-8 weeks for a passport application to be processed (although, for a higher fee, you can have the passport expedited and cut the time down to as little as 8 days.) My recommendation, if you're even thinking of walking a camino in the next year or two, apply now for the passport. They're good for 10 years.
Depending on expedited services selected, the cost of a US passport as of 2020 is $110. See the fee schedule here: https://travel.state.gov/content/dam/passports/forms-fees/Passport%20Fees%20Chart_TSG.pdf
In addition to the passport, while traveling, it's often a good idea to use a travel wallet of some sort. These usually have a neck strap or waist belt that allow you to keep your passport (and other important papers, currency, and credit cards) on your person (even while sleeping).
In my experience, theft isn't a huge issue on the caminos, but it does happen. It's easy to replace just about anything on the camino, although passports pose a greater replacement problem. So, it's better to be safe.