How large is the seed in an encryption algorithm such as stream cipher?
The seed of a pseudorandom generator that is used as a stream cipher is called a key.
The most common key sizes are 128 bits and 256 bits. (That's symmetric keys, where there's no cheaper way to break than brute force. Asymmetric cryptography typically relies on keys having certain mathematical properties that make the algorithm work, but also enable better attacks than brute force.)
To understand how hard it is to break a 128-bit key, meaning that you have to try $p \cdot 2^{128}$ possibilities for a chance of success $p$, consider these orders of magnitude:

A fast processor might be able to try about one billion keys per second (1GHz).
A powerful attacker might be able to afford one billion processors.
The universe is about 1 billion billion seconds old (a little over 30 billion years).
1 billion is slightly less than $2^{30}$ ($2^{30} = (2^{10})^3 = 1024^3 \gtrsim (10^3)^3 = 10^9$).

So if you're extremely rich and willing to wait until the universe is twice as old, you can enumerate about $2^{120}$ keys. If you're only willing to wait 30 years, you can only enumerate about $2^{90}$ keys, giving you a chance of breaking any particular key of less than one in 250 billion. Conclusion: a 128-bit key is sufficient for security against brute force.
A 256-bit key provides a security margin that can cover:

Hypothetical advances in cryptanalysis that make the algorithm slightly easier to break.
Hypothetical advances in computation, such as quantum computers, which effectively halve the key size.
A very powerful adversary who tries to break basically every encrypted communication, and is interested in even succeeding for a tiny fraction.
A very powerful adversary who records basically every encrypted communication, and would detect if the same key is used twice, which could then let them discover the key by non-cryptographic means (exploiting a bug, demanding it at gunpoint, a subpoena, …). Due to the “birthday paradox”, there's a macroscopic chance that two instances will generate the same random key if the number of instances is at least about the square root of the number of possible keys, so 8 billion people using a billion different keys each (about $2^{63}$ keys in total) would be around the threshold for 128-bit keys.


In practice, key size just isn't an issue except in legacy software and protocols that are at least 2 decades out of date (keys shorter than 128 bits became obsolete when AES became the de facto standard in the early 2000s). Cryptography gets broken by non-cryptographic means or due to implementation weaknesses such as side channels, or gets circumvented altogether due to non-cryptography-related security vulnerabilities (binary exploits, escaping exploits, misconfigured controls, phishing, …). Cryptographic algorithms are usually the strongest part of information security.