A simple solution is to change your document class to one of the KOMA-script classes that corresponds to the document class you use (book=scrbook, report=scrreprt and article=scrartcl). Then, you may use KOMA-script’s build in commands to format the captions.
I have listed most of KOMA-script’s caption format commands in the MWE. The two commands that I assume will be of the most interest for you, is
\addtokomafont{caption}{\footnotesize}
\addtokomafont{captionlabel}{\usekomafont{caption}}
The first line adds attributes to the text part of the caption (\caption{<text>}), i.e. the text inside the curly brackets. The second adds font attributes to the label part, (Figure 1, Tabell 1, etc.) The command \usekomafont{<fontset>}) replicates all font attributes set for the text part to the label.
If you want to reset the attributes from scratch, use \setkomafont{caption}{<attributes>}, instead of\addtokomafont`. For example, if you want the label in italic, but not the text part, use
\addtokomafont{captionlabel}{\usekomafont{caption}\itshape}
Here is an MWE and the output:

\documentclass[UKenglish, demo]{scrartcl}
\usepackage{lmodern}
\usepackage{scrlayer-scrpage}
\usepackage[babel=true]{microtype}
\usepackage{babel}
\usepackage{graphicx}
\usepackage{booktabs, bigdelim, rotating}
\KOMAoptions{headings=small,%
captions=tableheading,%
}
% Section
\let\raggedsection\flushleft
% Caption and figures
\renewcommand*{\captionformat}{:\ }
\addtokomafont{caption}{\footnotesize}
\addtokomafont{captionlabel}{\usekomafont{caption}}
\setlength{\belowcaptionskip}{0.5\baselineskip}
\setlength{\abovecaptionskip}{0.5\baselineskip}
\setlength{\intextsep}{0.5\baselineskip}
\begin{document}
\title{Capital Asset Prices}
\author{W.\,T.\,F.\,Dull}
\maketitle
\section{Section}
\label{sec:intro}
\textsc{One of the problems} which has plagued the world is bad typesetting of figures and tables. Now we have KOMA-script to help us.
\begin{figure}[!h]
\centering
\includegraphics[width=0.5\columnwidth]{figure1.png}
\caption{A demo of figure captions\label{fig-1}}
\end{figure}
Even table captions look better with KOMA-script, but that is no surprise. The example is borrowed from another question posted at Stackexchange.com
\begin{table}[!htbp]
\caption{A demo of figure captions\label{fig-1}}
\centering
\begin{tabular}{llcc@{}}
\cmidrule[\heavyrulewidth](l){2-4}
& header1 & header 2 & header 3 \\
\cmidrule(l){2-4}
\ldelim\{{4}{4mm}[\parbox{4mm}{\rotatebox[origin=c]{90}{group1}}] & 1 & a & g \\
& 2 & b & h \\
& 3 & c & i \\
& 3 & c & i \\\addlinespace[0.75ex]
\ldelim\{{6}{4mm}[\parbox{4mm}{\rotatebox[origin=c]{90}{group2}}] & 4 & d & j \\
& 5 & e & k \\
& 6 & f & l \\
& 7 & g & m \\
& 8 & h & n \\
& 9 & i & o \\
\cmidrule[\heavyrulewidth](l){2-4}
\end{tabular}
\end{table}
\end{document}
\captionargument, specify the formatting of the text as it appears in the caption and list of figures as part of the document setup in the class file or preamble – David Carlisle Aug 14 '19 at 23:33\captionsetupcommand, or perhaps you are using a class that has built in features, or ... – David Carlisle Aug 14 '19 at 23:39captionpackage is not compatible with certain document classes; it is for this reason that various people have asked you for more information about the structure of your document, beginning with the\documentclassinstruction. – Mico Aug 15 '19 at 02:13