ArcMap will always project the data frame using the first layer you bring into the Table of Contents. Looking at your screenshot, it seems like your dataframe isn't actually projected, but is rather being drawn in the WGS84 Geographic Coordinate System (GCS WGS 84).
You have at least one layer that is projected using the NZGD_2000 Transverse Mercator projection, and a couple layers that are not projected (your economic zone and coastlines topo 150k).
Here's what I would do - I would start by projecting all of my source data to the same projected coordinate system. Then I would insert a new data frame into my ArcMap document. Bring all the layers you're interested in analyzing, and then finally add your basemap. ArcMap will project the basemap on-the-fly and you shouldn't see any warnings.
Please let me know if this is helpful information.
Edit - it would be worthwhile to familiarize yourself with the differences between geographic coordinate systems and projected coordinate systems. Namely why it is that you can't reliably measure distances using a GCS (hint - distances between degrees aren't uniform across the globe ).