How do we choose refractive index value in optical fiber design?
The difference between the two refractive index values determines the range of angles that can undergo total internal reflection, and thus the numerical aperture (range of angles that can propagate inside the fiber) of the fiber.
If you make the difference large, you get a high NA fiber, which is sometimes used for things like relaying LED light from a source to where it is needed.  However, as the NA increases, the V-number of the fiber goes up, which means that to remain single mode the core diameter must be made smaller and smaller.  If you want a multimode fiber, that is fine, and MM fibers usually have very high NA so that they can be used to relay things like LED light.
For lasers however, that is usually a bad thing.  Since each mode in the fiber moves at a different speed, if you try to use a MM fiber for communications, you'll find that your maximum fiber length is very short before your signals get scrambled.  Alternatively, you can make the core very small to keep the V-number low, but now you are concentrating your energy in a very small area, which means that nonlinear effects like self phase modulation go way up, which again limits how far you can send a signal down the fiber before it is destroyed by nonlinearity.
For a long range, high data rate fiber, you actually want the lowest NA you can get, which will mean a very large core and very low nonlinearity while still being single mode.  Fibers like this can be used to send signals under the ocean.  The challenge is that they cannot be bent as sharply (due to the very limited range of angles) and the need to have very precise manufacturing to ensure such a small (but constant) difference in refractive index.  In practice, typical single mode fibers are about an NA of 0.1, while very low nonlinearity fibers can be 0.06 or lower (and therefore a mode field diameter of tens of microns), although these are much more expensive.  At the other extreme, you have the thick plastic cables used for things like TOSLINK, where the distance is a few meters so you don't care that some modes are moving much faster than others.