How to repair the caulking around basement walkout doors and what to use?
It's hard to prescribe an exact product, but I ran into similar issues recently myself and here is what I learned. Some of this may be helpful in making a decision...
There used to be two main options for caulk: silicone and latex. These days there are 'siliconized' latex caulks that perform well and are easier to use than 100% silicone. Silicone is hard to clean up, and nothing much sticks to it.
Latex (a.k.a. acrylic) caulk, whether siliconized or not, can be cleaned up with water, and other things can be applied on top such as paint or additional layers of caulk. So there is less reason to go with 100% silicone anymore.
If the sealant needs to accommodate expansion and contraction, a caulk advertised as window/trim may be a good choice as these are designed to be flexible.
DAP DYNAFLEX 230 is one such siliconized flexible caulk that is generally considered to be of high quality.
If the existing base is well adhered and non-silicone, you can probably clean it and apply latex on top. If there's any doubt, it's better to remove it and start from a known sturdy substrate.
You can probably narrow the field by eliminating caulks advertised as 'non-shrinking', 'paintable', 'quick drying', as those don't matter so much in this case. Adhesion, water-sealing, and possibly elasticity, are the things I would prioritize. Good luck!