You can start with as little as $300-500 and succeed. Although if you have $1000+ as a starting budget than that allows you to really create a higher quality store and also have plenty of money to start running Facebook or Instagram ads, for example.
I've seen people start with $200 and build successful million dollar/year stores.
I've also seen people with $3000 start and completely fail.
Again, it all depends on how bad you want success - how bad you want freedom in your life. How bad you want this to work so that you can finally reach your destiny and start living a life true to yourself, while owning a business that you can be proud of.
With dropshipping, the good part is you don't need to order any inventory to start. You can continue dropshipping until you find a winning product, and at that point make the decision whether you want to order a few hundred units with your brand name on it or not.
Answer: You can start with as little as $300-500, and $1000+ sets you up for maximum success.