Um, both? After all, negative vs. positive rights is mostly a philosophical distinction, not a constitutional/legal one. This does not mean that there's anything "negative" about it; it's just a fancy way of saying the government can't do something because the Constitution bars it from doing so, like the First Amendment's protection of expression, religion, peaceful assembly, etc. While a "positive" right forces the government to do something it may not want to do. And the Constitution clearly includes both. Even the Bill of Rights (first ten Amendments to the Constitution) does, too.
It's true that most of the rights in the Constitution are negative rights. Example from the recent District Court decision on whether cruise ships can require documentation of vaccination: the Florida rule that cruise lines can't demand your vaccine record card, but can ask if you've been vaccinated, violates the First Amendment's express negative right (incorporated by the 14th Amendment against the States) that government "shall make no law respecting" speech. The First Amendment says government shall not compel (or block) speech based on its content, so if you have to look at what was said (or done) to know whether the law has been broken or not, it very likely will violate the First Amendment. The Florida rule prohibits information being presented in writing (ironically, by a government agency) while permitting the same content from being spoken, so was based on content and subject to a successful negative right challenge.
But then there is the Sixth Amendment's power to compel the government to provide criminal defense counsel and speedy trials, which is a positive right. And tons of things that have elements of both negative and positive (which tend to be in "penumbral" rights found in interpretations of language, rather than the common understanding of clear and specific constitutional language), like the right to privacy, marriage or nondiscrimination. As a result, in legal interpretation these days, the distinction is more often phrased whether the asserted right comes from the text of the Constitution or an interpretation of it; textual rights tend to be more persuasive to most judges, simply because they don't require as much research to see how they've been interpreted in the past. IOW, we argue more about "positive," non-textual rights because it isn't clear if they are just confirmation bias or normative (what the law "should be" instead of what it "is"), while it's pretty clear where negative "shall not" prohibitions start the constitutional journey. But under the Sixth Amendment, if you get arrested, you get rights against the government. It's right there in the Bill of Rights:
"In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury ..., and to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witnesses in his favor [IOW, subpoenas], and to have the Assistance of Counsel for his defence." Hence, Miranda warnings when you're arrested, to be sure you know your rights. (Yes, I know Ernesto Miranda himself was supposedly a scumbag but the genius of the system is that even scumbags get rights. Both negative and positive.)