How does finding a bigger prime help or hinder cryptography? And with many forms of cryptography (e.g. DH kex, ECC) primes only enter as the modulus of the mod

How does finding a bigger prime help or hinder cryptography? And with many forms of cryptography (e.g. DH kex, ECC) primes only enter as the modulus of the modular arithmetic where being larger is usually not particularly helpful.