Which joint along a staircase can you seal?
Start by naming the two parts. A fixed stringer against a painted wall, a floor meeting the bottom step and a loose tread are three different jobs. Sealant cannot hold a tread in place.

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Walk the staircase before opening a cartridge
Look from below and again from the landing. Photograph the full run, not just the widest spot. The gap may taper to nothing halfway up. Walk normally and note movement; do not force a loose part to open the crack. Have a rocking tread or loose riser assessed by the installer.
Before opening the cartridge, try the Presch silicone gun in the tightest corner. Check you can turn your wrist and release pressure at the end of the run.
Choose the finish for each edge
At a stable stringer-to-wall joint, a compatible paintable acrylic may suit limited movement in a dry room. For a floating floor, find its installation drawing first. Preserve the expansion clearance unless the system permits an elastic joint there. Never use a finishing bead to glue a renovation tread to the wall.
At a stable, dry stringer-to-wall joint, trial OTTOSEAL A205 with the intended paint. It is not an adhesive for a moving tread.
Dry-fit before finishing
Look at a physical colour sample beside the floor in daylight and in the shadow beneath the first step. Try the profiling tool in the inside corner before dispensing. Protect the step from foot traffic while the approved joint cures. If access is tight, agree the order of work with the flooring installer and painter.
If the floor system explicitly allows an elastic joint at the foot of the stairs, compare the colour of OTTOSEAL S125 with a physical sample. This is a link to the sealant, not a colour chart.
Before opening the cartridge, try the FUGENFUX at the inside corner between the stringer and skirting. Choose the profile that fits without reducing contact at the edges.
Tools and materials for this job

OTTOSEAL A205
For a dry joint that will be painted. Check movement and paint compatibility.

OTTOSEAL S125
For a compatible floor joint where sealing is allowed. This is the sealant, not a colour chart.

Presch silicone gun
Manual cartridge gun. Check you can release pressure in the last centimetre.

FUGENFUX
Profiling tool. Dry-fit the edge that reaches both sides of the joint.
Can I fill every gap between the treads?
Only where the stair design permits it. A widening gap, creaking joint or lifting tread needs investigation. At a floating floor the correct finish may be a trim that leaves the clearance free.