it depends on many things....
1) do you want to target the English-speaking market in general and / or the USA + Canada market in particular? Because if you do, you would need a bigger budget than if you targeted a 3rd world country / your own 3rd world country;
2) if it's the English-speaking market you're after you then need to think about the niche. Niches like finances, medical, or web hosting will cost you a ton of money spent on quality content (think upwards of $40-50 per 1000 words article, and you need longer than 1000 words anyway, and a lot of content in general + a lot of strong backlinks which could cost you upwards of $500-1000 per month).
I don't know whether these numbers are accurate, but even as estimates they can still put things in perspective for you if the niche you've found for your blog is in one of these 3 super-competitive niches that only good SEOers and / or big-budgeted entities get into, and stay on top of the rankings.
3) regardless of which niche and country you're targeting you also need money for several crucial things, like...
- domain (usually around $10 per year);
- hosting (usually between $50 and $200 per year, depending on which hosting you choose)
- content (if you can't write the content yourself you will need to outsource, and even decent quality content which you might still need to fact-check and proofread could cost you $7-8 per 1000 words, whereas superior quality content could cost you anywhere from $12 to $25 or more per 1000 words. And, like all content-oriented websites such as blogs, you are going to need at least a few articles each week until your blog takes off, which probably means a few $100s per month spent on content alone)
- SEO tools in general, and keyword / niche research tools in particular and (optional) backlinking tools and services to build backlinks to your site and monitor them, as well as monitor your rankings. You will most likely spend at least $50-$100 per month in such tools and services, as well. And this only if you find some good deals and discounts for these tools / services.
So, as you can see your budget depends on many things, and at the very minimum you should have at least $200-300 per month set aside for your SEO / blogging journey, but probably more if your niche is tough to break into and maintain your rankings in, or if your goals are high and you target tough keywords.