Jumping spiders are mature when they have mature sex organs - fully modified pedipalps in males and a visible epigynum in females. (More on that later). SIZE IS NOT A GOOD INDICATOR.
In some species, the sexual dimorphism is apparent in young spiders, such as in Phidippus regius (Regal Jumping Spiders). The females will have lighter colored hairy faces and hair over their chelicerae, where as the males are dark with bald faces. The orange/tan varieties will have tan and orange peeking through at very young ages/small sizes, while the males remain stark black and white. However, there are some P. regius males that are grey with hairy faces and can be confusing.
In most species, males and females look identical until maturity at their last molt.
The best way to tell if a jumping spider is male or female, other than color with species that show dimorphism early, is waiting until sub-adult stage. The males will have bulged pedipalps, though not fully modified. The spiders of the same clutch, or locale, that are the size of, or larger than, their sub-adult male clutchmates with thin palps are assumed females.
Fully modified male pedipalps look like boxing gloves or old-fashioned telephone receivers. The last joint of the pedipalp are thickened with a dip in the middle.
Sub-adult male pedipalps are rounded and thickened, but the dip is not so pronounced.
This sub-adult P. otiosus Canopy Jumping Spider male (below), is nearly identical to females of the species, except for the rounded bulging sub-adult male pedipalps.
It's easiest to spot thickened sub-adult male palps and adult male palps on a 3/4th view. However, you can also see the chunky thickened sub-adult palps from the front if you look carefully/closely past the backlit hair.
Comparatively, female pedipalps remain very thin all the way to the tips. The pedipalps may appear to flare when viewed from the bottom, but it is only the very tip. People often mistake skinny females for males, especially in P. audax Bold Jumping Spiders, where females are also black and relatively hairless, like the males. But, proportions can be misleading there. Always look at the pedipalps and confirm with presence of an epigynum in mature females.
Notice how this female's (below) pedipalps flare at the very tips, but there is no bulging compared to her male counterparts.
Here's a female P. otiosus that looks very male due to her dark coloring. But the pedipalps and epyginum reveals that she is actually a female.
A mature female will have an epigynum that can be visible from underneath. It's easiest to spot if you put them in a clear container and use a phone to magnify the details. You're looking for a shiny indent/puckered hole like structure right in the middle top of the abdomen, between the book lungs.
Here's what an epigynum looks like on a dark female. Again, note the "flared" pedipalps, accentuated by the angle of the photo, but the palps are thin through the last joint, hence female palps.
Males and immature spiders may have two indents where an epigynum would be. The spider pictured below is actually a sub-adult male.
WHY SIZE DOESN'T MATTER AND CAN'T BE USED TO DETERMINE MATURITY OR SEX OF A JUMPING SPIDER
I see the mistake being made often of assuming a spider is one sex or another, or mature/immature, due to its size. Size can vary vastly between individuals as well as spiders of the same species from different locales.
Usually people assume males are smaller than females, but there are definitely males that are also larger than the females of the same species. It is just not an accurate indicator. Sizes can also differ drastically as the examples below show. The smaller spiders are P. otiosus from Ohio. The larger spiders are P. otiosus from Georgia. They are the same species, but the size difference of the different locales are drastic, especially in the males.