Where's the rest of your party? The fighters on the Holmes and Moldvay basic sets brought their wizard friends, and that was against smaller dragons. A wizard would be a big help here. Don't try to tell me that your wizard is off-screen. That's a weaselly lie, and I would not expect it of a noble barbarian like you.
WTF, Mentzer Red Box fighter? According to the bestiary in the Red Box, that Red Dragon has 10 Hit Dice at least - probably more, based on how big it is and how many ewers he has. But let's say it only has 45 HP. How much damage do you do? 1d8+6, max, if you have 18 Strength and a +3 sword?
Let's say you hit every round, and kill him in five rounds. How much damage is it doing to you in the meantime? Its breath weapon does damage EQUAL TO ITS HIT POINTS. Even if you make your saving throw, you take 22 HP of damage on round 1 of combat. And next round it's going to be a claw/claw/bite, which is not much better. Shouldn't you tackle some more appropriate solo opponent from the Red Box, like a Crystal Statue or a Thoul?
What's your Armor Class anyway, Mentzer Red Box Fighter? Don't try to tell me that that's a plate mail mankini you're wearing. That's chain mail - if you're LUCKY. Even with a shield, and 18 Dex to go with your 18 Strength, that makes your AC what, 1? That dragon is hitting you more than half of the time. Over the course of four combat rounds, you're probably getting tagged with at least 12d8 worth of claw/claw/bite. That's 54 average damage, for a total of 76 so far.
WTF, Mentzer Red Box Fighter? How many HP do you have anyway? Don't tell me you have an 18 Con to go with your 18 Strength and 18 Dex?? Even if you rolled all 5's on your d8 Hit Dice, you'd need to be at least level 11 to survive 76 damage (after level 9, you only get 2 HP per level).
OK, so you might beat the dragon. If you have 18 Str, Con, and Dex; a +3 magic sword; you make your saving throw vs breath weapon; and you never miss an attack over five rounds. AND IF YOU'RE LEVEL 11. In which case... what are you doing on the D&D box for characters level 1-3? WTF?