What would budget headphones would you recommend for someone who is starting to get into mixing?
Truth is, you’ll adapt to whatever headphones you have as long as they are decent and you learn them well.
As YourMan said, as long as they are decent, you should get used to how they sound. Be aware of how they “colour” the mixes and tracks so you know not to mix too heavily in one direction or another and always check your mix on multiple players.
That said, for studio work, AKG and Sennheiser are good brands with many options for different budgets. However, I have some Marshall headphones for personal/enjoyment listening and they sound really good,. Pretty flat response and not over coloured and the mixes cut through really well.